Definitions matrices y determinantes fula 2010 english subir

ESCUELA DE INGENIERÍA DE PETROLEOS



The matrices are first apparent by the year 1850, introduced by JJ Sylvester The initial development of the theory is due to the mathematical WR Hamilton in 1853.

In 1858, A. Cayley introduced matrix notation as shorthand to write a system of m linear equations with n unknowns.

The matrices are used in the numerical computation in solving systems of linear equations, differential equations and partial derivatives.


The use of matrices (arrays) is now an essential part of programming languages, since most data are entered into the computers at the tables arranged in rows and columns: spreadsheets, databases, etc..

Besides its usefulness for the study of systems of linear equations, matrices appear naturally in geometry, statistics, economics, computer science, physics, etc ...


An matriz is a set of items of any nature, but in general, numbers are usually arranged in rows and columns.

Order matrix is called "m × n" to a set of elements aij rectangular arranged in m rows and n columns. The order of a matrix is also called business size, where m and n natural numbers.


The matrices are denoted with capital letters: A, B, C, ... and the elements of the same small letters and subscripts indicating the place occupied: a, b, c, ...

A generic element occupying the ith row and column is written j aij. If the generic element in brackets also represents the entire a matriz:

A = (aij)


When we refer either to rows or columns speak of lines. The total number of elements of a matrix is m × n Am ° N. In mathematics, both the list and the tables are called generic matrices.

A numerical list is a set of numbers arranged one after the other.

SAME MATRICES

Two matrices A = (aij) m × n and B = (bij) p × q are equal if and only if, in the same places with the same elements, namely:


The study of array types is made based primarily on two criteria:

1. Depending on the arrangement of its elements:Triangular MatricesMatrix transposeSymmetric matrixAnti-symmetric matrix

2. ACCORDING TO THE BEHAVIOR OF MATRIX OPERATIONS TO CERTAINOrthogonal matrixIdempotentNilpotent matrixUnipotent matrix


TRIANGULAR MATRICES

DEFINITION (triangular matrices).

A matrix of order n is called upper triangular if all entries below the main diagonal are zeros and is called lower triangular if all entries above the main diagonal are zeros. A triangular matrix is called upper and lower diagonal matrix. A diagonal matrix in which all the main diagonal entries are equal is called a scalar matrix.

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TRIANGULAR MATRICES

Definition: upper triangular matrix

The matrix A = (aij) a square matrix of order n. We say that A is upper triangular if all elements of A, located under the main diagonal are zero, ie aij = 0 for all i> j, i, j = 1 ,...., n. For example the matrices

are upper triangular matrices.


Definition: Lower Triangular Matrix

The matrix A = (aij) a square matrix of order n. We say that A is lower triangular if all elements of A located above the main diagonal are zero, ie aij = 0 for all i <j, i, j = 1 ,...., n. For example, arrays

are lower triangular.


Definition: Diagonal matrix

The matrix A = (aij) a square matrix of order n. We say that A is a diagonal matrix if upper triangular and lower triangular, is aij = 0 for all i different from j, i, j = 1 ,...., n. For example, arrays

are diagonal matrices.


Consider the upper triangular matrices A, B of order four, as defined in the previous examples, we see that

1) A + B is upper triangularView that A + B obtains

which again is upper triangular.


2) a.A. is upper triangular (for all i R)

If we make

obtain an upper triangular matrix

3) A.B. is upper triangular.

If we calculate with this product DERIVE

which again is upper triangular.


4) If A has an inverse, A-1 is upper triangular.

We take the matrix A, which is upper triangular, we will calculate its inverse. Consider a general matrix of order 4

the inverse of A candidate must verify the matrix equation

that gives us the total simplification of equations that has to verify


SOLVE to get the elements of the inverse:

therefore the inverse of A, if the IA will be called

is upper triangular.


B) MATRIX Transpose

Matrix Transpose

Let a matrix A = any (aij) mxn order. We say that the matrix B = (bij) of order nxm is the transpose of A if the rows of A are the columns of B. This operation is usually denoted by At = A '= B.

In DERIVE this operation is implemented in the kernel. To calculate the matrix transpose of a matrix A defined simply edit DERIVE "A`. " For example, if


for its transpose simply edit the expression DERIVE

(Note that a 'is the grave accent), to simplify expression gives the transpose of A:

As can be seen is a very simple operation.


SOME TYPES OF MATRICES

There are some matrices that appear frequently and according to its form, its elements, ... different names:







Para establecer las reglas que rigen el cálculo con matrices se desarrolla un álgebra semejante al álgebra ordinaria, pero en lugar de operar con números lo hacemos con matrices.


C) SIMMETRIC MATRICES

Symmetric matrices:

We say that a matrix A of order mxn is a matrix chasm © trical if it coincides with its transpose, ie A = A `

Is evident that the symmetric matrices must be square matrices.


D) BANDEADED MATRIX

In mathematics, particularly in the theory of matrices, a matrix is banded sparse matrix whose nonzero elements are confined or limited to a diagonal band: understanding the main diagonal and zero or more diagonal sides.

Formally, an n * n matrix A = a (i, j) is a banded matrix if all elements of the matrix are zero outside the diagonal band whose rank is determined by the constants K1 and K2:

Ai, j = 0 if j <i - K1 j> i + K2, K1, K2 ≥ 0.


The quantities k1 and bandwidth k2son left and right, respectively. The bandwidth of the matrix is k1 + k2 +1 (in other words, the smallest number of adjacent diagonals of which are non-zero elements).

A banded matrix with k1 = k2 = 0ES a diagonal matrix, a banded matrix with k1 = k2 = 1 is a tridiagonal matrix, when k1 = k2 = 2 one has a matrix pentadiagonal and so on. If you put k1 = 0, k2 = n-1 gives the definition of an upper triangular matrix, similarly to k1 = n-1, k2 = 0 one obtains a lower triangular matrix.


EXAMPLES

1. Diagonal matrices.

For a matrix of order 3. k1 = k2 = 0 then Ai, j = 0 if j i <i and j>

The above matrix has a bandwidth of 1 and is given the special name of diagonal matrix.


EXAMPLES

2. Tridiagonal matrices.

Eg for a matrix of order 4. k1 = k2 = 1 then Ai, j = 0 if j i +1 <i-1 and j>

The above matrix has a bandwidth of 3 and is given the special name tridiagonal matrix.


E) Matrix Transpose

If we have a matrix (A) any order mxn, then its transpose is another array (A) of order nxm where they exchange the rows and columns of the matrix (A). The transpose of a matrix is denoted by the symbol "T" and is, therefore, that the transpose of the matrix A is represented by AT. Clearly, if A is an array of size mxn, At its transpose will nxm size as the number of columns becomes row and vice versa.

If the matrix A is square, its transpose is the same size.

Properties:a) ie the transpose of the transpose is the initial matrix.b)c)


Examples:

1.

Then the matrix transpose of A is

2. If

Then the matrix transpose of A is


F) MATRIX INVERSE

Inverse matrix is called a square matrix W and represent the A-1, a matrix that verifies the following property: A-1 ° A = A ° A-1 = I

We say that a square matrix is "regular" if its determinant is nonzero, and is "unique" if its determinant is zero.


PROPIEDADES :

"There's only inverse of a square matrix if it is regular. "The inverse of a square matrix, if it exists, is unique.

"Between arrays there is NO division operation, the inverse matrix performs similar functions.


SUM OF MATRICES

The sum of two matrices A = (aij) m × n and B = (bij) p × q of the same size (equidimensional): m = p and n = q is another matrix C = A + B = (cij) m × n = (aij + bij)


Es una ley de composición interna con las siguientes PROPIEDADES: ·Asociativa: A+(B+C) = (A+B)+C·Conmutativa : A+B = B+A·Elemento. neutro: ( matriz cero 0m×n ) , 0+A = A+0 = A·Elemento. simétrico : ( matriz opuesta -A ) , A + (-A) = (-A) + A = 0 Al conjunto de las matrices de dimensión m×n cuyos elementos son números reales lo vamos a representar por Mm×n y como hemos visto, por cumplir las propiedades anteriores, ( M, + ) es un grupo abeliano. ¡¡La suma y diferencia de dos matrices NO está definida si sus dimensiones son distintas. !!


PRODUCT NUMBER FOR A REAL MATRIX

To multiply a scalar by a scalar matrix is multiplied by all elements of the matrix, obtaining another matrix of the same order.


It is a law of composition with the following external

PROPERTIES:


PRODUCT OF MATRICES

Given two matrices A = (aij) m × n and B = (bij) where n p × q = p, ie the number of columns in the first matrix equals the number of rows of the matrix B is defined Product A · B as follows:

The element takes the place (i, j) in the product matrix is obtained by adding the products of each element in row i of matrix A by the corresponding column j of matrix B.


The determinant is a function that assigns to a matrix of order n, a single real number called the determinant of the matrix. If A is a matrix of order n, the determinant of the matrix A we denote by det (A) or also (the bars do not mean absolute value).

DEFINITION

(Determinant of a matrix of order 1)

If a matrix of order one, then det (A) = a.


DEFINITION (Minors and cofactors of a matrix of order n):

Let A be a matrix of order, we define the least associated with the item of A as the determinant of the matrix obtained by deleting row i and column j of the matrix A. The cofactor associated with the element of A is given by


DEFINITION (Determinant of a matrix of higher order)

If A is a matrix of order , then the determinant of the matrix A is the sum of the elements of the first row of A multiplied by their respective cofactors.




RULE SARRUS

Step 1: Enter the matrix A and then the first two columns of A as follows

Step 2: Calculate the products indicated by arrows (which are listed below.) The products covered by the arrows directed downward are taken with positive sign, while the products corresponding to the arrows that go up are taken with negative sign.

+ + + - - -


RULE SARRUS

Step 3: Add the products with the appropriate signs as determined in step 2:

COMMENT

Sarrus rule can be used only for determinants of order 3.



http://personal.redestb.es/ztt/tem/t6_matrices.htm

http://docencia.udea.edu.co/GeometriaVectorial/uni2/seccion21.html Steven C. Chapra, “Methods Numeric's for Engineering” Quinta

Edition. Mac Graw Hill.

Stewart, James. "Calculus, Early Transcendent." 4 ed. Tr. Andrew Sesti. Mexico, Ed Thomson, 2002. p. 1151

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Definitions matrices y determinantes fula 2010 english subir