Input-output modelExplain Input Output Analysis as an application of Matrices. Explain the concept and application using an example.From Wikipedia, the free encyclopediaJump to: navigation, search

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This article is about the economic model. For the computer interface, see Input/output.

In economics, an input-output model uses a matrix representation of a nation's (or a region's) economy to predict the effect of changes in one industry on others and by consumers, government, and foreign suppliers on the economy. Wassily Leontief (1905-1999) is credited with the development of this analysis. Francois Quesnay developed a cruder version of this technique called Tableau économique. Leontief won the Nobel Memorial Prize in Economic Sciences for his development of this model. And, in essence, Léon Walras's work Elements of Pure Economics on general equilibrium theory is both a forerunner and generalization of Leontief's seminal concept. Leontief's contribution was that he was able to simplify Walras's piece so that it could be implemented empirically. The International Input-Output Association[1] is dedicated to advancing knowledge in the field of input-output study, which includes "improvements in basic data, theoretical insights and modelling, and applications, both traditional and novel, of input-output techniques."

Input-output depicts inter-industry relations of an economy. It shows how the output of one industry is an input to each other industry. Leontief put forward the display of this information in the form of a matrix. A given input is typically enumerated in the column of an industry and its outputs are enumerated in its corresponding row. This format, therefore, shows how dependent each industry is on all others in the economy both as customer of their outputs and as supplier of their inputs. Each column of the input-output matrix reports the monetary value of an industry's inputs and each row represents the value of an industry's outputs. Suppose there are three industries. Column 1 reports the value of inputs to Industry 1 from Industries 1, 2, and 3. Columns 2 and 3 do the same for those industries. Row 1 reports the value of outputs from Industry 1 to Industries 1, 2, and 3. Rows 2 and 3 do the same for the other industries.

While most uses of the input-output analysis focuses on the matrix set of interindustry exchanges, the actual focus of the analysis from the perspective of most national statistical agencies, which produce the tables, is the benchmarking of gross domestic product. Input-output tables therefore are an instrumental part of national accounts. As suggested above, the core input-output table reports only intermediate goods and services that are exchanged among industries. But an array of row vectors, typically aligned below this matrix, record non-industrial inputs by industry like payments for labor; indirect business taxes; dividends, interest, and rents; capital consumption allowances

(depreciation); other property-type income (like profits); and purchases from foreign suppliers (imports). At a national level, although excluding the imports, when summed this is called "gross product originating" or "gross domestic product by industry." Another array of column vectors is called "final demand" or "gross product product consumed." This displays columns of spending by households, governments, changes in industry stocks, and industries on investment, as well as net exports. (See also Gross domestic product.) In any case, by employing the results of an economic census which asks for the sales, payrolls, and material/equipment/service input of each establishment, statistical agencies back into estimates of industry-level profits and investments using the input-output matrix as a sort of double-accounting framework.

The mathematics of input-output economics is straightforward, but the data requirements are enormous because the expenditures and revenues of each branch of economic activity have to be represented. As a result, not all countries collect the required data and data quality varies, even though a set of standards for the data's collection has been set out by the United Nations through its System of National Accounts[2](SNA): the replacement for the current 1993 SNA standard is pending.when? Because the data collection and preparation process for the input-output accounts is necessarily labor and computer intensive, input-output tables are often published long after the year in which the data were collected--typically as much as 5-7 years after. Moreover, the economic "snapshot" that the benchmark version of the tables provides of the economy's cross-section is typically taken only once every few years, at best, although many developed countries estimate input-output accounts annually and with much greater recency.

Contents

[hide] 1 Usefulness 2 Key Ideas 3 Forecasting and/or Analysis Using Input-Output 4 Input-output Analysis Versus Consistency Analysis 5 See also 6 Bibliography

7 External links

[edit] Usefulness

In addition to studying the structure of national economies, input-output economics has been used to study regional economies within a nation, and as a tool for national and regional economic planning. Indeed a main use of input-output analysis is for measuring the economic impacts of events as well as public investments or programs as shown by IMPLAN and RIMS-II. But it is also used to identify economically related industry clusters and also so-called "key" or "target" industries--industries that are most likely to enhance the internal coherence of a specified economy. By linking industrial output to

satellite accounts articulating energy use, effluent production, space needs, and so on, input-output analysts have extended the approaches application to a wide variety of uses.

[edit] Key Ideas

Leontief's text remains one of the best expositions of input-output analysis. Nonetheless, two books--a rather fundamental one by William Miernyk[3] and another by Ronald E. Miller and Peter D. Blair--probably have greater international currency. The latter is presently being rewritten and re-released, this time by Cambridge University Press.

Consider the production of the ith sector. We may isolate (1) the quantity of that production that goes to final demand,ci, (2) to total output, xi, and (3) flows xij from that industry to other industries. We may write a transactions tableau

Table: Transactions in a Three Sector EconomyEconomic Activities

Inputs to Agriculture

Inputs to Manufacturing

Inputs to Transport

Final Demand

Total Output

Agriculture 5 15 2 68 90Manufacturing 10 20 10 40 80Transportation 10 15 5 0 30Labor 25 30 5 0 60

or

Note that in the example given we have no input flows from the industries to 'Labor'.

We know very little about production functions because all we have are numbers representing transactions in a particular instance (single points on the production functions):

The neoclassical production function is an explicit function

Q = f(K,L),

where Q = Quantity, K = Capital, L = Labor,

and the partial derivatives ( ) are the demand schedules for input factors.

Leontief, the innovator of input-output analysis, uses a special production function which depends linearly on the total output variables xi. Using Leontief coefficients aij, we may manipulate our transactions information into what is known as an input-output table:

or

Now

gives

Rewriting finally yields

Introducing matrix notation, we can see how a solution may be obtained. Let

denote the total output vector, the final demand vector, the unit matrix and the input-output matrix, respectively. Then:

provided (I − A) is invertible.

There are many interesting aspects of the Leontief system, and there is an extensive literature. There is the Hawkins-Simon Condition on producibility. There has been interest in disaggregation to clustered inter-industry flows, and the study of constellations of industries. A great deal of empirical work has been done to identify coefficients, and data have been published for the national economy as well as for regions. This has been a healthy, exciting area for work by economists because the Leontief system can be extended to a model of general equilibrium; it offers a method of decomposing work done at a macro level.

Transportation is implicit in the notion of inter-industry flows. It is explicitly recognized when transportation is identified as an industry – how much is purchased from transportation in order to produce. But this is not very satisfactory because transportation requirements differ, depending on industry locations and capacity constraints on regional production. Also, the receiver of goods generally pays freight cost, and often transportation data are lost because transportation costs are treated as part of the cost of the goods.

Walter Isard and his student, Leon Moses, were quick to see the spatial economy and transportation implications of input-output, and began work in this area in the 1950s developing a concept of interregional input-output. Take a one region versus the world case. We wish to know something about interregional commodity flows, so introduce a column into the table headed “exports” and we introduce an “import” row.

Table: Adding Export And Import Transactions

Economic Activities 1 2 … … Z Exports Final Demand Total Outputs

1

2

…

…

Z

Imports

A more satisfactory way to proceed would be to tie regions together at the industry level. That is, we could identify both intra-region inter-industry transactions and inter-region inter-industry transactions. The problem here is that the table grows quickly.

Input-output is conceptually simple. Its extension to a model of equilibrium in the national economy is also relatively simple and attractive but requires great skill and high-quality data. One who wishes to do work with input-output systems must deal skillfully with industry classification, data estimation, and inverting very large, ill-conditioned matrices. Moreover, changes in relative prices are not readily handled by this modeling approach alone. Of course, input-output accounts are part and parcel to a more flexible form of modeling, Computable general equilibrium models.

Two additional difficulties are of interest in transportation work. There is the question of substituting one input for another, and there is the question about the stability of coefficients as production increases or decreases. These are intertwined questions. They have to do with the nature of regional production functions.

[edit] Forecasting and/or Analysis Using Input-Output

Table: Interregional Transactions

Economic Activities

AgNorth Mfg

... ... AgEast Mfg

... ... AgWest Mfg

... ... ExportsTotal Outputs

North Mfg

...

...

Ag

East Mfg

...

...

Ag

West Mfg

...

...

Table: Input-Output Model for Hypothetical Economy Total requirements from regional industries per dollar of output delivered to final demand

Purchasing Industry Agriculture Transport Manufacturer Services

Selling Industry

Agriculture 1.14 0.22 0.13 0.12

Transportation 0.19 1.10 0.16 0.07

Manufacturing 0.16 0.16 1.16 0.06

Services 0.08 0.05 0.08 1.09

Total 1.57 1.53 1.53 1.34

[edit] Input-output Analysis Versus Consistency Analysis

Despite the clear ability of the input-output model to depict and analyze the dependence of one industry or sector on another, Leontief and others never managed to introduce the full spectrum of dependency relations in a market economy. In 2003, Mohammad Gani[4], a pupil of Leontief, introduced Consistency Analysis in his book 'Foundations of Economic Science' (ISBN 984320655X), which formally looks exactly like the input-output table but explores the dependency relations in terms of payments and intermediation relations. Consistency analysis explores the consistency of plans of buyers and sellers by decomposing the input-output table into four matrices, each for a different kind of means of payment. It integrates micro and macroeconomics in one model and deals with money in an ideology-free manner. It deals with the flow of funds via the movement of goods.

[edit] See also

Computable general equilibrium

Economic base analysis Gross Output Industrial organization IPO Model Net output Shift-share analysis

[edit] Bibliography

Dietzenbacher, Erik and Michael L. Lahr, eds. Wassilly Leontief and Input-Output Economics. Cambridge University Press, 2004.

Isard, Walter et al. Methods of Regional Analysis: An Introduction to Regional Science. MIT Press 1960.

Lahr, Michael L. and Erik Dietzenbacher, eds. Input-Output Analysis: Frontiers and Extensions. Palgrave, 2001.

Leontief, Wassily W. Input-Output Economics. 2nd ed., New York: Oxford University Press, 1986.

Miller, Ronald E. and Peter D. Blair. Input-Output Analysis: Foundations and Extensions. Prentice Hall, 1985.

Miller, Ronald E., Karen R. Polenske, and Adam Z. Rose, eds. Frontiers of Input-Output Analysis. N.Y.: Oxford UP, 1989.[HB142 F76 1989/ Suzz]

Miernyk, William H. The Elements of Input-Output Anaysis, 1965.[5]. Polenske, Karen. Advances in Input-Output Analysis. 1976. ten Raa, Thijs. The Economics of Input-Output Analysis. Cambridge University

Press, 2005. US Department of Commerce, Bureau of Economic Analysis . Regional

multipliers: A user handbook for regional input-output modeling system (RIMS II). Third edition. Washington, D.C.: U.S. Government Printing Office. 1997.

San José State UniversityDepartment of Economics

applet-magic.comThayer WatkinsSilicon Valley

& Tornado AlleyUSA

Input-Output Analysis and Related Methods

IntroductionHawkins-Simon

ConditionDynamics

Input-output analysis is one of a set of related methods which show how the parts of a system are affected by a change in one part of that system. Input-output analysis specifically shows how industries are linked together through supplying inputs for the output of an economy. Suppose there are only two industries producing Coal and Steel. Coal is required to produce steel and some amount of steel in the form of tools is required to produce coal. Suppose the input requirements per ton output of the two products are:

Industry Coal Steel

Coal 0 3

Steel 0.1 0

Suppose it desired that the Coal industry produce a net output of 200,000 tons of coal and the Steel industry a net ouputput of 50,000 tons. If the Coal industry just produces 200,000 tons and the Steel industry produces 50,000 tons some of outputs are used up in producing the other output. To produce 50,000 tons of Steel requires 3(50,000)=150,000 tons of coal. Likewise the production of 200,000 tons of coal requires (0.1)(200,000)=20,000 tons of steel. The net outputs of coal and steel would then be 200,000-150,000=50,000 tons of coal and 50,000-20,000=30,000 tons of steel. In other words, in order to get net outputs of 50,000 tons of coal and 30,000 tons of steel it is necessary to produce 200,000 tons of coal and 50,000 tons of steel. But we want net outputs of 200,000 tons of coal and 50,000 tons of steel. We would at least have to produce an additional amount to replace the coal and steel used up in producing 200,000 tons of coal and 50,000 tons of steel. Those amounts, as we saw above, are 150,000 tons of coal and 20,000 tons of steel. But in producing these amount we will also use up coal and steel. In fact, we will use up 3(20,000)=60,000 tons of coal and (0.1)(150,000)=15,000 tons of steel. So we must increase production again to cover these amounts.

We can think of these production increases as Rounds of production. In Round 1 we produce the net outputs we are trying to achieve. In Round 2 we produce the outputs that were used up in producing Round 1 outputs. Then in Round 3 we produce the outputs used up in producing Round 2, and so on. The figures are given in the following table.

Round Coal Steel

1 200,000 50,000

2 150,000 20,000

3 60,000 15,000

4 45,000 6,000

5 18,000 4,500

6 13,500 1,800

7 5,400 1,350

8 4,050 540

9 1,620 405

The totals for the first nine rounds are 497,570 tons of coal and 99,595 tons of steel. This approximate how much we must produce to achieve the net outputs we seek. The exact amounts, found by more advanced methods, are 500,000 tons of coal and 100,000 tons of steel. We see that 3(100,000)=300,000 tons of coal are used up in producing the 100,000 tons of steel leaving 500,000-300,000=200,000 tons of coal as net output. Also (0.1)(500,000)=50,000 tons of steel are used up in producing the 500,000 tons of coal leaving us with 100,000-50,000=50,000 tons of steel as net output.

We find the exact figures by solving two algebraic equations. If x1 is the output of coal and x2 is the output of steel, then the conditions that have to be satisfied are:

x1 - 3x2 = 200,000 x2 - 0.1x1 = 50,000.  

This is a set of two equations in two unknows and we can solve it using simple algebra.

A systematic way to solve for any set of net outputs is to find how much coal and steel are needed to produce a net output of one ton of coal. The equations which have to be satisfied are:

x1 - 3x2 = 1 x2 - 0.1x1 = 0.  

The solution is x1=1.42857 and x2=0.14286. To get the amounts necessary to produce a net output of 200,000 tons we multiply these figures by 200,000. The outputs required are 285,714 tons of coal and 28,571 tons of steel.

For a net output of one ton of steel the equations to be satisfied are:

x1 - 3x2 = 0 x2 - 0.1x1 = 1.  

The solution is x1=4.28571 and x2=1.42857. To get the amounts necessary to produce a net output of 50,000 tons of steel we multiply these figures by 50,000. We get an output of coal of 214,286 tons and 71,429 tons of steel. The total outputs required for 200,000 tons of coal and 50,000 tons of steel are then

285,714 + 214,286 = 500,000 tons of coal and   28,571 + 71,429 = 100,000 tons of steel.  

The outputs of coal and steel required to achieve a net output of one ton of coal may be considered the direct and indirect requirements for one ton of coal. Likewise for the outputs necessary for a net output of one ton of steel. These can be put together in a table to contrast them with the direct requirements of production:

Direct Requirements

Industry Coal Steel

Coal 0 3

Steel 0.1 0

Direct and Indirect Requirements

Industry Coal Steel

Coal 1.42857 4.28571

Steel 0.14286 1.42857

The about table is the one that gives the most information about how the industries are inter-relataed.

The direct and indirect requirements are usually determined using matrix operations. A matrix is simply a rectangular array of numbers; i.e., a table. A matrix is characterized by the number of rows and columns. An n by m matrix, written n×m, is a matrix with n rows and m columns. A matrix with only one column is also called a column vector. If a matrix has only one row it is usally called a row vector.

The element of matrix A that is in the i-th row and j-th column is denoted as Ai,j. The addition of two matrices A and B with the same number of rows and columns is defined as the matrix C such that Ci,j=Ai,j+Bi,j. Subtraction of matrices is defined in an analogous manner.

The definition of multiplication of matrices that is useful is not the multiplication of corresponding elements. Instead multiplication is defined only for matrices having the proper number of rows and columns; i.e., the number of rows of the second matrix has to be equal to the number of columns of the first matrix. If A is an n×m matrix and B is a m×p matrix then their product AB is defined to be an n×p matrix C such that

Ci,j = ΣAi,kBk,j  

where the summation is over k=1 to k=m.

The levels of production of the Coal and Steel industries can be represented as a 2×1 matrix (a column vector) X where

X = | x1 |    | x2 |

 

Likewise the levels of net output required to be met, usually called final demands, f1 and f2 can be represented as a column vector (2×1 matrix) F where

F = | f1 |    | f2 |

 

The direct requirements per units of output for an economy with two industries can be represent as a 2×2 matrix A; i.e.,

A = | A1,1   A1,2 |    | A2,1   A2,2 |

 

For the Coal/Steel economy above

A = |   0   3 |    | 0.1   0 |

 

The amounts of production used up in producing x1 and x2 are equal to

A1,1x1 + A1,2x2 A2,1x1 + A2,2x2  

This is just the product of the matrix A and the matrix X; i.e., AX. The production levels are given by X so the net productions after the amounts used up in production are deducted are given by

X - AX  

We want this to be equal to the required levels F so the equations to be satisfied are, in matrix form,

X - AX = F  

At this point it is necessary to note that there are special type of square matrices, called identity matrices and denoted as I, that consist of 1's on the diagonal that runs from the upper left to the lower right and 0's everywhere else. The 2×2 identity matrix is

I = | 1   0 |    | 0   1 |

 

The virtue of the identity matrices is that the product of an identity matrix with any other matrix for which the product is defined is just the other matrix. In particular, IX is just X. It turns out that it is often useful to represent a matrix as a product with the identity matrix. The equations to be satisfied by the levels of output are, in matrix form,

X - AX = F which can be expressed asIX - AX = F  

The advantage of this second representation is that it can be factored; i.e.,

IX - AX = (I-A)X so the matrix equations to be satisfied is (I-A)X = F  

For the Coal/Steel economy above

I-A = |  1   -3 |      | -0.1   1 |

 

The solution to this equation would involve carrying out some algebraic operation so we end up with X being equal to some matrix. Generally the problem we face is that we have a matrix equation BX=C and we want to end up with X=something. Suppose that we could find a special matrix D such that DB=I. Then we could multiply both sides of the matrix equality BX=C to get DBX=DC. Since DB=I this means we would have IX=DC, but IX is the same as X so we have the solution to the equations as X=DC. The special matrix D such that DB=I is called the inverse of B and it is denoted as B-1. So the solution to the matrix equation BX=C is X=B-1C. The only problem in finding a solution to BX=C is finding B-1.

At this point we do not know even if such a matrix exists. It turns out that there is a simple test to determine whether an inverse exists. The test is based upon the determinant of the matrix. If the determinant is not equal to zero then an inverse exists and if the determinant is equal to zero then an inverse does not exist. For a 2×2 matrix B the determinant of B is

det(B) = B1,1B2,2 - B1,2B2,1  

For a 2×2 matrix the determination of the inverse is very simple but it is not as simple for higher order matrices. to get the inverse of a 2×2 matrix B we just interchange B1,1 and B2,2 and change the signs of B1,2 and B2,1 and then divide all of the elements of the resulting matrix by det(B); i.e.,

B-1 = | B2,2/det(B)   -B1,2/det(B) |    | -B2,1/det(B)    B1,1/det(B) |

 

For the Coal/Steel economy above the (I-A) was

I-A = | 1   -3 |       | -0.1   1 |

 

so it determinant is (1)(1)-(-3)(-0.1)=1-0.3=0.7. Thus it does have an inverse and that inverse is

(I-A)-1 = (1/0.7)| 1     3 |                  | 0.1     1 |

 

or, carrying out the indicated division,

(I-A)-1 = |  1.4286     4.2857 |    | 0.1429     1.4286 |

 

When we multiply this matrix times the vector of final demands F, where

F = | 200,000 |    |  50,000 |

 

we get the solution

X = | 500,000 |    | 100,000 |

 

In conclusion, the direct and indirect requirements per unit of final demands are given by the columns of the inverse of the matrix I-A; i.e., (I-A)-1.

For the Coal/Steel economy the inverse of (I-A)

(I-A)-1 = |  1.4286   4.2857 |    | 0.1429     1.4286 |

 

tells us that for each ton of final demand for coal the economy has to produce 1.4286 tons of coal and 0.1429 tons of steel. For each ton of final demand for steel the economy has to produce 4.2857 tons of coal and 1.4286 tons of steel.

Interregional and International InteractionUsing the Methods of Input-Output Analysis

Input-Output Analysis arose to deal with the problem of interindustry demand, but the same method can be used to show how changes in one region affect the economies of regions linked to it. Suppose we have information on how changes in production in Santa Clara and Santa Cruz Counties affect the demand for each other's output. (Santa Clara County is essentially the famed Silicon Valley and Santa Cruz County is a county to the south of it over the Santa Cruz Mountains and on Monterey Bay of the Pacific Ocean.) If production in Santa Clara County increases there will be more income not only for residents of Santa Clara County but also for the residents of Santa Cruz County because some of the jobs in Santa Clara County will go to Santa Cruz County residents. The residents of both counties will decide how much of their income they will spend, where, and for what. Some of that spending will be in the two counties and be for goods and services that are produced locally. Likewise when production in Santa Cruz County increases some of the jobs will go to Santa Clara County residents and some of these will spend their income in Santa Cruz County as well as in Santa Clara County. Suppose we have that information in matrix form:

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 0.5 0.1

Santa Cruz 0.2 0.4

This is like a matrix of marginal propensities to consume in macroeconomic theory. In macroeconomic theory if income is spent for products outside of the economy it is considered a leakage. In this regional setting some of these leakages leak back into the economy.

The matrix above corresponds to the matrix A in input-output analysis.

In macroeconomic theory the income multiplier k is equal to:

1/(1-c)  

where c is the marginal propensity to consume.

This could be written as:

k = (1-c)-1.  

With the regional interaction there is a matrix of multipliers and the matrix is equal to:

(I-A)-1.  

For the above matrix the matrix I-A is:

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 0.5 -0.1

Santa Cruz -0.2 0.6

The determinant of I-A is (0.5)(0.6)-(-0.1)(-0.2)=0.30-0.02=0.28.

This means the matrix I-A does have an inverse. Remember that for a 2x2 matrix the inverse is found by interchanging the diagonal elements and changing the sign of the off-diagonal elements, then dividing every element by the determinant. This gives:

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 2.14286 0.35714

Santa Cruz 0.71429 1.78571

This means that when the demand for Santa Clara County's output increases by $1 the output in Santa Clara County increases by $2.14 and in Santa Cruz County by $0.71. On the other hand, if the demand for Santa Cruz County's output increases by $1 then output in Santa Cruz County increases by $1.79 and in Santa Clara County by $0.36.

The above shows how once the matrix A is known how the inter-relationships between the parts is determined in the form of the inverse of the (I-A) matrix. So once A is determined the rest is merely numerical computation. But the matrix A first has to be established. The derivation of the matrix A involves several economic processes. First, there is the distribution of income (and jobs) to the subregions. This is given in the form of a matrix which will be called the matrix J (for jobs). Suppose J has the following value:

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 0.75 0.20

Santa Cruz 0.25 0.80

This says that 75% of the jobs and income go to Santa Clara County residents and 25% go to residents of Santa Cruz County. On the other hand, 20% of the jobs and income in Santa Cruz County go to Santa Clara County residents and the other 80% to Santa Cruz County residents.

But not all of a dollar of production goes for labor income. Let us say that in both counties one third of the revenue goes for labor income. This means that the effect of additional dollars of production would have the following effects on incomes. This is the matrix Y (for income).

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 0.250 0.067

Santa Cruz 0.083 0.267

There is also the matrix that tells where people spend their money and how much of it goes for local production. This is the matrix S (for spending).

  County of Residence

County ofSpending

Santa Clara Santa Cruz

Santa Clara 0.80 0.30

Santa Cruz 0.10 0.60

This says that that when Santa Clara residents get another dollar of income 80% is spent in Santa Clara County and another 10% is spent in Santa Cruz County. On the other hand, when Santa Cruz residents get another dollar of income 30% is spent in Santa Clara County and 60% at home in Santa Cruz County. In both cases all of the spending goes for goods or services which are produced in the county of the spending.

Note that the orientation of this table is opposite of the previous tables.

To construct the matrix A we have to follow a dollar of production to its dispursement as income to the two counties and the allocation of the recipients spending between the two counties. This illustration is going to leave out several other important economic processes such how much of labor income goes for taxes, savings and imports. These omissions are to keep the detail to a minimum.

According to matrix J, when a dollar of production is produced in Santa Clara County, $0.25 goes to Santa Clara County residents who spend 80% of it in Santa Clara County and $0.083 goes to Santa Cruz County residents who spend 30% of it in Santa Clara County. Altogether then the $1 of production in Santa Clara County leads to (0.25)(.8)+(0.083)(.3)=0.225 of addition consumer demand in Santa Clara County. This is the element in the first row, first column of the matrix A. The dollar of additional production also leads to increased demand in Santa Cruz County; i.e,. (0.25)(0.1)+(0.083)(.6)=0.075. This is the element in the second row, first column of A.

When an additional dollar of production takes place in Santa Cruz County the additional spending in Santa Clara County is (0.063)(0.8)+(0.267)(.3)=0.131, the element of the A matrix in the first row, second column. The final element is the

spending in Santa Cruz County resulting from an additional dollar of production in Santa Cruz County. This is (0.63)(0.1)+(0.267)(0.6)=0.167. Thus the A matrix is

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 0.225 0.131

Santa Cruz 0.075 0.167

The (I-A) is then

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 0.775 -0.131

Santa Cruz -0.075 0.834

The determinant of this matrix is 0.6365, so there is an inverse. That inverse is

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 1.3102 0.2058

Santa Cruz 0.1178 1.2175

This says that when there is an additional dollar of demand in Santa Clara County production goes up by $1.31 in Santa Clara County and about $0.12 in Santa Cruz County. On the other hand, when demand in Santa Cruz County increase by one dollar production in Sanat Clara County increases by about $0.21 and about $1.22

in Santa Cruz County. But these are increases in sales and production rather than income.

The increases in income are found by determining the proportion of production going to income (one third in this example) and then distributing it according to the J matrix. The result of this computation for Santa Clara County income due to an increase in Santa Clara production is (1/3)(1.3102)(0.75)+(1/3)(0.1178)(0.20)=0.335. The results of the computation is then

  County of Production

County ofResidence

Santa Clara Santa Cruz

Santa Clara 0.335 0.133

Santa Cruz 0.141 0.342

Matrix (mathematics)From Wikipedia, the free encyclopediaJump to: navigation, search

Specific entries of a matrix are often referenced by using pairs of subscripts.

In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, such as

An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general. Matrices consisting of only one column or row define the components of vectors, while higher-dimensional (e.g., three-dimensional) arrays of numbers define the components of a generalization of a vector called a tensor. Matrices with entries in other fields or rings are also studied.

Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations, which are higher-dimensional analogs of linear functions of the form f(x) = cx, where c is a constant; matrix multiplication corresponds to composition of linear transformations. Matrices can also keep track of the coefficients in a system of linear equations. For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.

Matrices find many applications. Physics makes use of matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail matrices with an infinite number of rows and columns. Graph theory uses matrices to keep track of distances between pairs of vertices in a graph. Computer graphics uses matrices to project 3-dimensional space onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to matrices. The latter is a recurring need in solving ordinary differential equations. Serialism and dodecaphonism are musical movements of the 20th century that use a square mathematical matrix to determine the pattern of music intervals.

A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old but still an active area of research. Matrix decomposition methods simplify computations, both theoretically and practically. For sparse matrices, specifically tailored algorithms can provide speedups; such matrices arise in the finite element method, for example.

[edit] Definition

A matrix is a rectangular arrangement of numbers.[1] For example,

An alternative notation uses large parentheses instead of box brackets:

The horizontal and vertical lines in a matrix are called rows and columns, respectively. The numbers in the matrix are called its entries or its elements. To specify a matrix's size, a matrix with m rows and n columns is called an m-by-n matrix or m × n matrix, while m and n are called its dimensions. The above is a 4-by-3 matrix.

A matrix with one row (a 1 × n matrix) is called a row vector, and a matrix with one column (an m × 1 matrix) is called a column vector. Any row or column of a matrix determines a row or column vector, obtained by removing all other rows respectively columns from the matrix. For example, the row vector for the third row of the above matrix A is

When a row or column of a matrix is interpreted as a value, this refers to the corresponding row or column vector. For instance one may say that two different rows of a matrix are equal, meaning they determine the same row vector. In some cases the value of a row or column should be interpreted just as a sequence of values (an element of Rn if entries are real numbers) rather than as a matrix, for instance when saying that the rows of a matrix are equal to the corresponding columns of its transpose matrix.

Most of this article focuses on real and complex matrices, i.e., matrices whose entries are real or complex numbers. More general types of entries are discussed below.

[edit] Notation

The specifics of matrices notation varies widely, with some prevailing trends. Matrices are usually denoted using upper-case letters, while the corresponding lower-case letters, with two subscript indices, represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other variables. An alternative notation involves the use of a double-underline with the variable name, with or without

boldface style, (e.g., ).

The entry that lies in the i-th row and the j-th column of a matrix is typically referred to as the i,j, (i,j), or (i,j)th entry of the matrix. For example, the (2,3) entry of the above matrix A is 7. The (i, j)th entry of a matrix A is most commonly written as ai,j. Alternative notations for that entry are A[i,j] or Ai,j.

Sometimes a matrix is referred to by giving a formula for its (i,j)th entry, often with double parenthesis around the formula for the entry, for example, if the (i,j)th entry of A were given by aij, A would be denoted ((aij)).

An asterisk is commonly used to refer to whole rows or columns in a matrix. For example, ai,∗ refers to the ith row of A, and a∗,j refers to the jth column of A. The set of all m-by-n matrices is denoted (m, n).

A common shorthand is

A = [ai,j]i=1,...,m; j=1,...,n or more briefly A = [ai,j]m×n

to define an m × n matrix A. Usually the entries ai,j are defined separately for all integers 1 ≤ i ≤ m and 1 ≤ j ≤ n. They can however sometimes be given by one formula; for example the 3-by-4 matrix

can alternatively be specified by A = [i − j]i=1,2,3; j=1,...,4, or simply A = ((i-j)), where the size of the matrix is understood.

Some programming languages start the numbering of rows and columns at zero, in which case the entries of an m-by-n matrix are indexed by 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1.[2] This article follows the more common convention in mathematical writing where enumeration starts from 1.

[edit] Interpretation as a parallelogram

The vectors represented by a matrix can be thought of as the sides of a unit square transformed into a parallelogram. The area of this parallelogram is the absolute value of the determinant of the matrix.

If A is a 2×2 matrix

then the matrix A can be viewed as the transform of the unit square into a parallelogram with vertices at (0,0), (a,b), (a + c, b + d), and (c,d). The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product xTAT, where x is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors

and in turn. These row vectors define the vertices of the unit square. With the more common matrix-vector product Ax, the parallelogram has

vertices at and (note that Ax = (xTAT)T ).

Further, the area of this parallelogram can be viewed as the absolute value of the determinant of the matrix A. When the determinant is equal to one, then the matrix represents an equi-areal mapping.

[edit] Basic operations

Main articles: Matrix addition, Scalar multiplication, Transpose, and Row operations

There are a number of operations that can be applied to modify matrices called matrix addition, scalar multiplication and transposition.[3] These form the basic techniques to deal with matrices.

Operation Definition ExampleAddition The sum

A+B of two m-by-n matrices A and B is calculated entrywise:

(A + B)i,j = Ai,j

+ Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.

Scalar multiplication

The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:

(cA)i,j = c · Ai,j.

Transpose The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:

(AT)

i,j = Aj,i.

Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e. the matrix sum does not depend on the order of the summands: A + B = B + A.[4] The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A.

Row operations are ways to change matrices. There are three types of row operations: row switching, that is interchanging two rows of a matrix, row multiplication, multiplying all entries of a row by a non-zero constant and finally row addition which means adding a multiple of a row to another row. These row operations are used in a number of ways including solving linear equations and finding inverses.

[edit] Matrix multiplication, linear equations and linear transformations

Main article: Matrix multiplication

Schematic depiction of the matrix product AB of two matrices A and B.

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot-product of the corresponding row of A and the corresponding column of B:

where 1 ≤ i ≤ m and 1 ≤ j ≤ p.[5] For example (the underlined entry 1 in the product is calculated as the product 1 · 1 + 0 · 1 + 2 · 0 = 1):

Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined.[6] The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ≠ k. Even if both products are defined, they need not be equal, i.e. generally one has

AB ≠ BA,

i.e., matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is:

whereas

The identity matrix In of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.

It is called identity matrix because multiplication with it leaves a matrix unchanged: MIn = ImM = M for any m-by-n matrix M.

Besides the ordinary matrix multiplication just described, there exist other less frequently used operations on matrices that can be considered forms of multiplication, such as the Hadamard product and the Kronecker product.[7] They arise in solving matrix equations such as the Sylvester equation.

[edit] Linear equationsMain articles: Linear equation and System of linear equations

A particular case of matrix multiplication is tightly linked to linear equations: if x designates a column vector (i.e. n×1-matrix) of n variables x1, x2, ..., xn, and A is an m-by-n matrix, then the matrix equation

Ax = b,

where b is some m×1-column vector, is equivalent to the system of linear equations

A1,1x1 + A1,2x2 + ... + A1,nxn = b1

...Am,1x1 + Am,2x2 + ... + Am,nxn = bm .[8]

This way, matrices can be used to compactly write and deal with multiple linear equations, i.e. systems of linear equations.

[edit] Linear transformationsMain articles: Linear transformation and Transformation matrix

Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real m-by-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm. Conversely, each linear transformation f: Rn → Rm arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f(ej), where ej = (0,...,0,1,0,...,0) is the unit vector with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f.

The following table shows a number of 2-by-2 matrices with the associated linear maps of R2. The blue original is mapped to the green grid and shapes, the origin (0,0) is marked with a black point.

Horizontal shear with m=1.25.

Horizontal flipSqueeze

mapping with r=3/2

Scaling by a factor of 3/2

Rotation by π/6R = 30°

Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps:[9] if a k-by-m matrix B represents another linear map g : Rm → Rk, then the composition g ∘ f is represented by BA since

(g ∘ f)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x.

The last equality follows from the above-mentioned associativity of matrix multiplication.

The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors.[10] Equivalently it is the dimension of the image of the linear map represented by A.[11] The rank-nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix.[12]

[edit] Square matrices

A square matrix is a matrix which has the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. A square matrix A is called invertible or non-singular if there exists a matrix B such that

AB = In.[13]

This is equivalent to BA = In.[14] Moreover, if B exists, it is unique and is called the inverse matrix of A, denoted A−1.

The entries Ai,i form the main diagonal of a matrix. The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While, as mentioned above, matrix multiplication is not commutative, the trace of the product of two matrices is independent of the order of the factors: tr(AB) = tr(BA).[15]

Also, the trace of a matrix is equal to that of its transpose, i.e. tr(A) = tr(AT).

If all entries outside the main diagonal are zero, A is called a diagonal matrix. If only all entries above (below) the main diagonal are zero, A is called a lower triangular matrix (upper triangular matrix, respectively). For example, if n = 3, they look like

(diagonal), (lower) and (upper triangular matrix).

[edit] DeterminantMain article: Determinant

A linear transformation on R2 given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one.

The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2-by-2 matrices is given by

the determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalises these two formulae to all dimensions.[16]

The determinant of a product of square matrices equals the product of their determinants: det(AB) = det(A) · det(B).[17] Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.[18] Using these

operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices.[19] This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.[20]

[edit] Eigenvalues and eigenvectorsMain article: Eigenvalues and eigenvectors

A number λ and a non-zero vector v satisfying

Av = λv

are called an eigenvalue and an eigenvector of A, respectively.[nb 1][21] The number λ is an eigenvalue of an n×n-matrix A if and only if A−λIn is not invertible, which is equivalent to

[22]

The function pA(t) = det(A−tI) is called the characteristic polynomial of A, its degree is n. Therefore pA(t) has at most n different roots, i.e., eigenvalues of the matrix.[23] They may be complex even if the entries of A are real. According to the Cayley-Hamilton theorem, pA(A) = 0, that is to say, the characteristic polynomial applied to the matrix itself yields the zero matrix.

[edit] Symmetry

A square matrix A that is equal to its transpose, i.e. A = AT, is a symmetric matrix. If instead, A was equal to the negative of its transpose, i.e. A = −AT, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfy A∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, i.e. the transpose of the complex conjugate of A.

By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; i.e., every vector is expressible as a linear combination of eigenvectors. In both cases, all eigenvalues are real.[24] This theorem can be generalized to infinite-dimensional situations related to matrices with infinitely many rows and columns, see below.

[edit] DefinitenessMatrix A; definiteness; associated quadratic form QA(x,y);

set of vectors (x,y) such that QA(x,y)=1

positive definite indefinite

1/4 x2 + y2 1/4 x2 − 1/4 y2

Ellipse Hyperbola

A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors x ∈ Rn the associated quadratic form given by

Q(x) = xTAx

takes only positive values (respectively only negative values; both some negative and some positive values).[25] If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.[26] The table at the right shows two possibilities for 2-by-2 matrices.

Allowing as input two different vectors instead yields the bilinear form associated to A:

BA (x, y) = xTAy.[27]

[edit] Computational aspects

In addition to theoretical knowledge of properties of matrices and their relation to other fields, it is important for practical purposes to perform matrix calculations effectively and precisely. The domain studying these matters is called numerical linear algebra.[28] As with other numerical situations, two main aspects are the complexity of algorithms and their numerical stability. Many problems can be solved by both direct algorithms or iterative approaches. For example, finding eigenvectors can be done by finding a sequence of vectors xn converging to an eigenvector when n tends to infinity.[29]

Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, e.g. multiplication of matrices. For example, calculating the matrix product of two n-by-n matrix using the definition given above needs n3 multiplications, since for any of the n2 entries of the product, n multiplications are necessary. The Strassen algorithm outperforms this "naive" algorithm; it needs only n2.807 multiplications.[30] A refined approach also incorporates specific features of the computing devices.

In many practical situations additional information about the matrices involved is known. An important case are sparse matrices, i.e. matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.[31]

An algorithm is, roughly speaking, numerically stable, if little deviations (such as rounding errors) do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace's formula (Adj (A) denotes the adjugate matrix of A)

A−1 = Adj(A) / det(A)

may lead to significant rounding errors if the determinant of the matrix is very small. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix' inverse.[32]

Although most computer languages are not designed with commands or libraries for matrices, as early as the 1970s, some engineering desktop computers such as the HP 9830 had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices.[33]

[edit] Matrix decomposition methodsMain articles: Matrix decomposition, Matrix diagonalization, and Gaussian elimination

There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix transformation or matrix decomposition techniques. The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.

The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U).[34] Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form.[35] Both methods proceed by multiplying the matrix by suitable elementary matrices, which

correspond to permuting rows or columns and adding multiples of one row to another row. Singular value decomposition expresses any matrix A as a product UDV∗, where U and V are unitary matrices and D is a diagonal matrix.

A matrix in Jordan normal form. The grey blocks are called Jordan blocks.

The eigendecomposition or diagonalization expresses A as a product VDV−1, where D is a diagonal matrix and V is a suitable invertible matrix.[36] If A can be written in this form, it is called diagonalizable. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ1 to λn of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right.[37] Given the eigendecomposition, the nth power of A (i.e. n-fold iterated matrix multiplication) can be calculated via

An = (VDV−1)n = VDV−1VDV−1...VDV−1 = VDnV−1

and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponential eA, a need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices.[38] To avoid numerically ill-conditioned situations, further algorithms such as the Schur decomposition can be employed.[39]

[edit] Abstract algebraic aspects and generalizations

Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors, which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional array of numbers.[40] Matrices, subject to certain requirements tend to form groups known as matrix groups.

[edit] Matrices with more general entries

This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field, i.e. a set where addition, subtraction, multiplication and division operations are defined and well-behaved, may be used instead of R or C, for example rational numbers or finite fields. For example, coding theory makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the coefficients of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field (e.g., to view a real matrix as a complex matrix whose entries happen to be all real) then allows considering each square matrix to possess a full set of eigenvalues. Alternatively one can consider only matrices with entries in an algebraically closed field, such as C, from the outset.

More generally, abstract algebra makes great use of matrices with entries in a ring R.[41] Rings are a more general notion than fields in that no division operation exists. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(n, R) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R-module Rn.[42] If the ring R is commutative, i.e., its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible.[43] Matrices over superrings are called supermatrices.[44]

Matrices do not always have all their entries in the same ring - or even in any ring at all. One special but common case is block matrices, which may be considered as matrices whose entries themselves are matrices. The entries need not be quadratic matrices, and thus need not be members of any ordinary ring; but their sizes must fulfil certain compatibility conditions.

[edit] Relationship to linear maps

Linear maps Rn → Rm are equivalent to m-by-n matrices, as described above. More generally, any linear map f: V → W between finite-dimensional vector spaces can be described by a matrix A = (aij), after choosing bases v1, ..., vn of V, and w1, ..., wm of W (so n is the dimension of V and m is the dimension of W), which is such that

In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W; thus this relation uniquely determines the entries of the matrix A. Note that the matrix

depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices.[45] Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.[46]

[edit] Matrix groupsMain article: Matrix group

A group is a mathematical structure consisting of a set of objects together with a binary operation, i.e. an operation combining any two objects to a third, subject to certain requirements.[47] A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group.[nb 2][48] Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups.

Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups. For example, matrices with a given size and with a determinant of 1 form a subgroup of (i.e. a smaller group contained in) their general linear group, called a special linear group.[49] Orthogonal matrices, determined by the condition

MTM = I,

form the orthogonal group.[50] They are called orthogonal since the associated linear transformations of Rn preserve angles in the sense that the scalar product of two vectors is unchanged after applying M to them:

(Mv) · (Mw) = v · w.[51]

Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group.[52] General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory.[53]

[edit] Infinite matrices

It is also possible to consider matrices with infinitely many rows and/or columns[54] even if, being infinite objects, one cannot write down such matrices explicitly. All that matters is that for every element in the set indexing rows, and every element in the set indexing columns, there is a well-defined entry (these index sets need not even be subsets of the natural numbers). The basic operations of addition, subtraction, scalar multiplication and transposition can still be defined without problem; however matrix multiplication may involve infinite summations to define the resulting entries, and these are not defined in general.

If infinite matrices are used to describe linear maps, then only those matrices can be used all of whose columns have but a finite number of nonzero entries, for the following reason. For a matrix A to describe a linear map f: V→W, bases for both spaces must have been chosen; recall that by definition this means that every vector in the space can be written uniquely as a (finite) linear combination of basis vectors, so that written as a (column) vector v of coefficients, only finitely many entries vi are nonzero. Now the columns of A describe the images by f of individual basis vectors of V in the basis of W, which is only meaningful if these columns have only finitely many nonzero entries. There is no restriction on the rows of A however: in the product A·v there are only finitely many nonzero coefficients of v involved, so every one of its entries, even if it is given as an infinite sum of products, involves only finitely many nonzero terms and is therefore well defined. Moreover this amounts to forming a linear combination of the columns of A that effectively involves only finitely many of them, whence the result has only finitely many nonzero entries, because each of those columns do. One also sees that products of two matrices of the given type is well defined (provided as usual that the column-index and row-index sets match), is again of the same type, and corresponds to the composition of linear maps.

Infinite matrices can also be used to describe operators on Hilbert spaces, where convergence and continuity questions arise, which again results in certain constraints that have to be imposed. However, the explicit point of view of matrices tends to obfuscate the matter,[nb 3] and the abstract and more powerful tools of functional analysis can be used instead.

[edit] Empty matrices

An empty matrix is a matrix in which the number of rows or columns (or both) is zero.[55]

[56] An empty matrix has no entries but it still has a well defined number of rows and columns, which are needed for instance in the definition of the matrix product. Thus if A is the 3-by-0 matrix A and B is the 0-by-3 matrix B, then AB is the 3-by-3 zero matrix (corresponding to the null map from a 3-dimensional space V to itself obtained obtained as composition of the unique map f from V to a 0-dimensional space Z, followed by the zero map g from Z back to V), while BA is the 0-by-0 matrix (corresponding to the unique map from Z to itself obtained as composition ). There is no common notation for empty matrices but most computer algebra systems will allow creating them and computing with them. Note that the determinant of the 0-by-0 matrix is 1 (and not 0 as might seem more natural): the Leibniz formula produces this value as a sum over the unique permutation of the empty set, with an empty product as term; also the Laplace expansion for a 1-by-1 matrix makes clear that the value of the 0-by-0 minor should be taken to be 1. This value is also consistent with the fact that the identity map from any finite dimensional space to itself has determinant 1, a fact that is often used as a part of the characterization of determinants.

[edit] Applications

There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.[57] Text mining and automated thesaurus compilation makes use of document-term matrices such as tf-idf in order to keep track of frequencies of certain words in several documents.[58]

Complex numbers can be represented by particular real 2-by-2 matrices via

under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above. A similar interpretation is possible for quaternions.[59]

Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.[60] Computer graphics uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accomplish tasks such as projecting a three-dimensional object onto a two-dimensional screen, corresponding to a theoretical camera observation.[61] Matrices over a polynomial ring are important in the study of control theory.

Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix using in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method.

[edit] Graph theory

An undirected graph with adjacency matrix

The adjacency matrix of a finite graph is a basic notion of graph theory.[62] It saves which vertices of the graph are connected by an edge. Matrices containing just two different values (0 and 1 meaning for example "yes" and "no") are called logical matrices. The distance (or cost) matrix contains information about distances of the edges.[63] These concepts can be applied to websites connected hyperlinks or cities connected by roads etc., in which case (unless the road network is extremely dense) the matrices tend to be sparse, i.e. contain few nonzero entries. Therefore, specifically tailored matrix algorithms can be used in network theory.

[edit] Analysis and geometry

The Hessian matrix of a differentiable function ƒ: Rn → R consists of the second derivatives of ƒ with respect to the several coordinate directions, i.e.[64]

It encodes information about the local growth behaviour of the function: given a critical

point x = (x1, ..., xn), i.e., a point where the first partial derivatives of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices (see above).[65]

At the saddle point (x = 0, y = 0) (red) of the function f(x,−y) = x2 − y2, the Hessian matrix

is indefinite.

Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f: Rn → Rm. If f1, ..., fm denote the components of f, then the Jacobi matrix is defined as [66]

If n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem.[67]

Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question.[68]

The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.[69]

[edit] Probability theory and statistics

Two different Markov chains. The chart depicts the number of particles (of a total of 1000) in state "2". Both limiting values can be determined from the transition matrices,

which are given by (red) and (black).

Stochastic matrices are square matrices whose rows are probability vectors, i.e., whose entries sum up to one. Stochastic matrices are used to define Markov chains with finitely many states.[70] A row of the stochastic matrix gives the probability distribution for the next position of some particle which is currently in the state corresponding to the row. Properties of the Markov chain like absorbing states, i.e. states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.[71]

Statistics also makes use of matrices in many different forms.[72] Descriptive statistics is concerned with describing data sets, which can often be represented in matrix form, by reducing the amount of data. The covariance matrix encodes the mutual variance of several random variables.[73] Another technique using matrices are linear least squares, a

method that approximates a finite set of pairs (x1, y1), (x2, y2), ..., (xN, yN), by a linear function

yi ≈ axi + b, i = 1, ..., N

which can be formulated in terms of matrices, related to the singular value decomposition of matrices.[74]

Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.[75][76]

[edit] Symmetries and transformations in physicsFurther information: Symmetry in physics

Linear transformations and the associated symmetries play a key role in modern physics. For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions, which behave as spinors.[77] For the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices, which are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses.[78]

[edit] Linear combinations of quantum states

The first model of quantum mechanics (Heisenberg, 1925) represented the theory's operators by infinite-dimensional matrices acting on quantum states.[79] This is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates.[80]

Another matrix serves as a key tool for describing the scattering experiments which form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators, where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles.[81]

[edit] Normal modes

A general application of matrices in physics is to the description of linearly coupled harmonic systems. The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions. The best way to obtain solutions is to determine the system's eigenvectors, its normal modes, by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of molecules: the internal vibrations of systems consisting of mutually bound component atoms.[82] They are also needed for describing mechanical vibrations, and oscillations in electrical circuits.[83]

[edit] Geometrical optics

Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there will be two different kinds of matrices, viz. a refraction matrix describing de madharchod refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix will apply. The optical system consisting of a combination of lenses and/or reflective elements is simply described by the matrix resulting from the product of the components' matrices.[84]

[edit] Electronics

Traditional mesh analysis in electronics leads to a system of linear equations which can be described with a matrix.

The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage v1 and input current i1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v2 and output current i2 as its elements. Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22). Calculating a circuit now reduces to multiplying matrices.

[edit] History

Matrices have a long history of application in solving linear equations. The Chinese text The Nine Chapters on the Mathematical Art (Jiu Zhang Suan Shu), from between 300 BC and AD 200, is the first example of the use of matrix methods to solve simultaneous equations,[85] including the concept of determinants, over 1000 years before its

publication by the Japanese mathematician Seki in 1683[citation needed] and the German mathematician Leibniz in 1693. Cramer presented his rule in 1750.

Early matrix theory emphasized determinants more strongly than matrices and an independent matrix concept akin to the modern notion emerged only in 1858, with Cayley's Memoir on the theory of matrices.[86][87] The term "matrix" was coined by Sylvester, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices which derive from the original one by removing columns and rows. Etymologically, matrix derives from Latin mater (mother).[88]

The study of determinants sprang from several sources.[89] Number-theoretical problems led Gauss to relate coefficients of quadratic forms, i.e., expressions such as x2 + xy − 2y2, and linear maps in three dimensions to matrices. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative. Cauchy was the first to prove general statements about determinants, using as definition of the determinant of a matrix A = [ai,j] the following: replace the powers aj

k by ajk in the polynomial

where Π denotes the product of the indicated terms. He also showed, in 1829, that the eigenvalues of symmetric matrices are real.[90] Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above; Kronecker's Vorlesungen über die Theorie der Determinanten[91] and Weierstrass' Zur Determinantentheorie,[92] both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established.

Many theorems were first established for small matrices only, for example the Cayley-Hamilton theorem was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrices. Frobenius, working on bilinear forms, generalized the theorem to all dimensions (1898). Also at the end of the 19th century the Gauss-Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Jordan. In the early 20th century, matrices attained a central role in linear algebra.[93] partially due to their use in classification of the hypercomplex number systems of the previous century.

The inception of matrix mechanics by Heisenberg, Born and Jordan led to studying matrices with infinitely many rows and columns.[94] Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.

[edit] Other historical usages of the word "matrix" in mathematics

The word has been used in unusual ways by at least two authors of historical importance.

Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word matrix in the context of their Axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" (0 order) the function will be identical to its extension[disambiguation needed]:

"Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables. Then any possible function other than a matrix is derived from a matrix by means of generalization, i.e. by considering the proposition which asserts that the function in question is true with all possible values or with some value of one of the arguments, the other argument or arguments remaining undetermined".[95]

For example a function Φ(x, y) of two variables x and y can be reduced to a collection of functions of a single variable, e.g. y, by "considering" the function for all possible values of "individuals" ai substituted in place of variable x. And then the resulting collection of functions of the single variable y, i.e. a∀ i: Φ(ai, y), can be reduced to a "matrix" of values by "considering" the function for all possible values of "individuals" bi substituted in place of variable y:

b∀ j a∀ i: Φ(ai, bj).

Alfred Tarski in his 1946 Introduction to Logic used the word "matrix" synonymously with the notion of truth table as used in mathematical logic.[96] Introduction

Input-output matrix is constructed on the simple idea that goods and services produced by economic sectors should be registered in a table simultaneously by origin and by destination. Commodities are produced by economic sectors (e.g. cotton produced by agriculture) and they serve as inputs in other sectors in order to produce their final products also called outputs (e.g. manufacturing industry such as textile industry using cotton from agriculture as input to produce its own output, i.e. clothes in cotton). Better said, the purchase of agricultural output by manufacturing is for use as inputs in producing manufacturing output. Such purchases are part of what is known as intermediate demand, which term refers to inter-industry transactions, i.e. goods and services bought by firms from other firms and used up in current production (this corresponds to Domestic intermediate matrix – see first quadrant in Table 1). The outputs are delivered to the final demand sector that comprises purchases by individuals for consumption, by firms for investment (in fixed capital such as machines, buildings, etc.), by government, and by foreigners (exportations) – this corresponds to the fourth quadrant in Table 1 called Domestic investment matrix. The use of this terminology “final demand” simply indicates that purchases by this sector are not for the purpose of

use in production (Common and Stagl, 2005)[2]. In addition to intermediate inputs mentioned above, firms use primary inputs. Those are services which are not bought from other firms but from individuals: these services are known as factors of production. These refer to wages and salaries as payments for labour services, interests paid on borrowing, rent paid for the use of equipment, building and land, profits paid for the entrepreneurship that is the function of organizing and risk-taking (Common and Stagl, 2005). – this corresponds to the second quadrant in Table 1, which is called Imported intermediate products matrix. We are not going to detail the third and the fifth quadrant here since they are not essential to the global understanding of the methodology. For more details, look at the legend of Table 1 or go to (OECD, 2006) and read pp.7-9.

Table 1. Example of an Input-output Matrix for Belgium in the year 2000. Source : OECD (2006a)

Table 1 shows an example of a real input-output matrix for Belgium in the year 2000. The columns represent the destination of inputs, and the rows sum the output of a sector. As you can see, only the total outputs (last line called industry output) are shown, not the total inputs. This is because such data is not necessary since the total inputs equal the total outputs. Normally, this total appears in the last column of the table.

Practical use of Input-output tables

If we want to estimate without Input-output analysis, which additional inputs would be needed if the fishing sector increased its production by one unit, we would need to measure the following: i) first round, direct effects on the industries that supply the fishing sector with nets, boats, fishing rods etc; and ii) a range of secondary (indirect) effects, since these supplier industries themselves require additional inputs for their production, in order to meet the additional demand coming from the fishing sector production system.

Fortunately, input-output matrices offer a solution to solve such problems immediately taking into account both direct as well as indirect effects. This can be particularly appreciable for assessing economic impact (both ex-ante and ex-post) of policy changes. Environmental impact can even be analyzed if we add environmental data to classical input-output tables in order to build green input-output tables.

For instance, if a policy option scenario for marine pollution management (e.g. a tax on plastic industry resulting in higher plastic prices or a governmental subsidies to the production of material of substitution that are biodegradable) results in technical changes

or in changes in final demand for plastics (a valuable material particularly in construction, packaging and fishing gear applications), I-O analysis can help us to deduce the following (adapted from Leontief, 1974, 193-209 pp.)[3] :

the policy options impact on the total level of pollution by plastic microparticles in the sea

the amount of pollution reduction in a particular sector resulting from the implementation of a policy option

the total pollution resulting from the final demand (demand from households, …) for products of each sector. For instance, keeping the example of plastic production, this means that the I-O table can tell us : “from the total amount Y of plastic pollutant in the sea, X tones are linked to agriculture, industrial and services activities contributing directly or indirectly to the supply of agricultural products to households. This is interesting since it does not only take into account the amount of pollutants from the agriculture sector for the production of agricultural products, but it also encompasses pollutants from other sectors intervening in the production of agricultural products. That is important since the agriculture sector also needs industrial products and services to generate its production. The same can be calculated for the supply of industrial products and services to households.

the impact of policy options on production level in other sectors (and so on the economy)

the impact of policy options on total employment in the region or in a particular sector

the impact of policy options on prices of goods and services

Limits of the method

The input-output (I-O) analysis is not able to capture environmental measures with a small economic impact (on GDP, on production, on employment…at national or regional level) because of data are too aggregated. Therefore, I-O is only relevant for activities having a wide economic impact such as construction of large infrastructures (railways or motorways infrastructure), modification of port activities, implementation of environmental policies targeting a whole sector, subsector or a branch of economic activities, etc.

Nevertheless, I-O analysis could also be relevant for a package of several policy options, each having a relatively small impact, but whose sum results in a large impact on the regional economy.

Walter Hecq (2006a)[4] summarized several other limits of I-O analysis. They are mentioned below.

First of all, environmental measure might affect output prices. For instance, if a governmental policy make compulsory for oil companies to install an oil de-sulfuration system (that prevents acid rains), the cost of this de-pollution system will be reflected in

oil price. Hence, all products requiring oil in their fabrication process will see their price modified too (e.g.: outputs from agriculture, electricity, ferrous metals, etc.). And due to an increase in their price, the demand for each of these products will probably decrease. The modification of the demand due to price variation must be integrated into I-O models but this makes them heavier to handle because of numerous products and/or diverse response functions. In that case, dynamic model such as CGE might be more suitable.

Moreover, I-O matrices give a static vision of the economy making difficult projections possibilities. However, it is possible to build dynamic I-O matrices but this is a bit more complicated.

Another disadvantage is the impossibility of substitution between production factors (labors, technical capital, land) while environmental policies might precisely have a structural effect on the long run on that aspect. Let’s take the example of an environmental policy aiming at decreasing greenhouse gases emissions by promoting research and development in energy efficiency in households. Imagine the instrument of this policy would consist in public subsidies to universities for research in building insulation new technologies. Such a measure might lead to reduction of households energy consumption and so a reduction in natural gas extraction burnt in power plants for electricity generation. In that case, the production factor “land” in the form of a natural resource (natural gas) has been partly substituted by the production factor “labors” (development of human knowledge in new insulation techniques).

Furthermore, I-O tables are published by national and regional authorities with few years delay. For instance in Belgium in 2007, the last I-O table available dates from 2000. Through the delayed publishing of national I-O tables, factor relations within single sectors can be changed to a quite big extend. Such old data on the status of the economy might make I-O analysis for the subsequent years quite inaccurate. However there are techniques enabling to actualize too old I-O matrices.

The last limit we would like to highlight is the need of regionalization of national I-O tables. It can happen that you find only national I-O tables while you want to work at regional level (i.e. at a smaller spatial scale). In that case you will need to apply regionalization methods which add to the difficulties.