Kit · 12/10/2005 · C c b e three matrices. Then C = A + B is called the addition of matrices A and B if c ij = a + b for all i and j. Business mathematics, Matrix, 15 th b emer

Kit Tyabandha, PhD Department of Mathematis, Mahidol University

De�nition 1. Let A = fa

ij

g, B = fb

ij

g and C = f

ij

g be

three matries. Then

C = A+B

is alled the addition of the matries A and B if

ij

= a

ij

+ b

ij

for all i and j.

Business mathematis, Matrix, 15

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November 2005 {1{ From 5

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November 2005 , as of 10

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Deember, 2005


De�nition 2. LetA = (a

ij

) be anm�nmatrix andB = (b

kl

)

an n � p matrix. Then the produt AB is an m � p matrix

C = (

il

) where,

il

=

n

X

k=1

a

ik

b

kl

where 1 � i � m and 1 � l � p.


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De�nition 3. The expression obtained by eliminating the n

variables x

1

; : : : ; x

n

from n equations,

a

11

x

1

+ : : :+ a

1n

x

n

= 0

...

...

a

n1

x

1

+ : : :+ a

nn

x

n

= 0

9=

;

(1)

is alled the determinant of this system of equations, Equation

1. The determinant of matrix A denoted by various di�erent no-

tations, for example det(A), jAj,

P

(�a

1

b

2

3

� � �), D(a

1

b

2

3

� � �),

or ja

1

b

2

3

� � � j.


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Example 1. For a linear system of three variables, Equation

1 an be written as,

a

1

x+ a

2

y + a

3

z = 0

b

1

x+ b

2

y + b

3

z = 0

1

x+

2

y +

3

z = 0

9=

;

(2)

Eliminating x, y and z from Equation 2 gives us,

a

1

b

2

3

� a

1

b

3

2

+ a

3

b

1

2

� a

2

b

1

3

+ a

2

b

3

1

� a

3

b

2

1

= 0


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De�nition 4. A minor M

ij

of any matrix A is the determi-

nant of a redued matrix obtained by omitting the i

th

row and

the j

th

olumn of A.


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Theorem 1. Determinant an be determined by,

jA j =

k

X

i=1

a

ij

C

ij

where C

ij

is alled the ofator of a

ij

. The ofator C

ij

an also

be denoted as a

ij

, and,

C

ij

= (�1)

i+j

M

ij

where M

ij

is a minor of A.


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De�nition 5. Any pairwisely ordered pair in a permutation

p is alled a permutation inversion in p if i > j and p

i

< p

j

.


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Theorem 2. Determination of the determinant an also be

determined by,

jA j =

X

�

(�1)

I(�)

n

Y

i=1

a

i;�(i)

where � is a permutation whih ranges over all permutations of

f1; : : : ; ng, and I(�) is alled the inversion number of �.


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Theorem 3. Let a be a onstant and A an n � n matrix.

Then,

j aA j = a

n

jA j

j �A j = (�1)

n

jA j

jAB j = jA j jB j

j I j = j

AA

�1

j = jA j j

A

�1

j = 1

jA j =

1

j

A

�1

j


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De�nition 6. A funtion in two or more variables is said to

be multilinear if it is linear in eah variable separately.


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Theorem 4. Determinants of matrix are multilinear in rows

and olumns.


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Example 2. Consider an 3� 3 matrix,

A =

��

a

1

a

2

a

3

a

4

a

5

a

6

a

7

a

8

a

9

��

What Theorem 4 says about multilinearity of determinants is

the same as saying that,

jA j =

��

a

1

0 0

a

4

a

5

a

6

a

7

a

8

a

9

��

+

��

0 a

2

0

a

4

a

5

a

6

a

7

a

8

a

9

��

+

��

0 0 a

3

a

4

a

5

a

6

a

7

a

8

a

9

��

and

jA j =

��

a

1

a

2

a

3

0 a

5

a

6

0 a

8

a

9

��

+

��

0 a

2

a

3

a

4

a

5

a

6

0 a

8

a

9

��

+

��

0 a

2

a

3

0 a

5

a

6

a

7

a

8

a

9

��


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De�nition 7. A onformal mapping is a transformation that

preserves loal angle. The terms funtion, map and transfor-

mation are synonyms.


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De�nition 8. A similarity transformation is a onformal

mapping the transformation matrix of whih is,

A

0

� BAB

�1

Here A and A

0

are similar matries.


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Theorem 5. Similarity transformation does not hange the

determinant.

Proof. The proof for this is simply,

j

BAB

�1

j = jB j jA j j

B

�1

j = jB j jA j

1

jB j

= jA j

{


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Example 3.

j

B

�1

AB � �I

j = j

B

�1

AB �B

�1

�IB

j

= j

B

�1

(A� �I)B

j

= j

B

�1

j jA� �I j jB j

= jA� �I j


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De�nition 9. Let A be a square, n � n matrix. Then the

trae of A is,

Tr(A) =

n

X

i=1

a

ii


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De�nition 10. The transpose of a matrix

A = fa

ij

g

is

A

T

= fa

ji

g


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De�nition 11. The omplex onjugate of a matrix

A = fa

ij

g

is

�

A = f�a

ij

g

where �a = p� qi if a = p+ qi.


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De�nition 12. Let �(n) or �(x) be a positive funtion, and

let f(n) or f(x) be any funtion. Then f = O(�) if jf j < A�

for some onstant A and all values of n and x. Here O is alled

the big-O notation whih denotes asymptotiity. The notation

f = O(�) is read, `f is of order �'.


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Theorem 6. Some other properties of the determinant are,

jA j =

��

A

T

��

j

�

A j = jA j

j I + �A j = 1 + Tr(A) +O(�

2

); for � small


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Example 4. For a square matrix A,

a. swithing rows hanges the sign of the determinant

b. fatoring out salars from rows and olumns leaves the value

of the determinant unhanged

. adding rows and olumns together leaves the determinant's

value unhanged

d. multiplying a row by a onstant gives the same determi-

nant multiplied by

e. if a row or a olumn is zero, then the determinant is zero

f. if any two rows or olumns are equal, then the determinant

is zero


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Theorem 7. Some properties of matrix trae are,

Tr(A) = Tr

�

A

T

�

Tr(A+B) = Tr(A) + Tr(B)

Tr(�A) = �Tr(A)


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Problem 1. Prove that,

�

A

T

�

�1

=

�

A

�1

�

T


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Theorem 8.

(AB)

T

= B

T

A

T

Proof.

�

B

T

A

T

�

ij

=

�

b

T

�

ik

�

a

T

�

kj

= b

ki

a

jk

= a

jk

b

ki

= (AB)

ji

= (AB)

Tij

{


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De�nition 13. Let A be a square matrix. Then the inverse

of A, if it exists, is A

�1

suh that,

AA

�1

= I

Furthermore, A is said to be nonsingular or invertible if its

inverse exists, otherwise it is said to be singular.


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Example 5. For a 2� 2 matrix,

A =

�

a b

d

�

the inverse of A is,

A

�1

=

1

ad� b

�

d �b

� a

�


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If A is a 3� 3 matrix, then the inverse of A is,

A

�1

=

1

jA j

�

det

�

m

ij

�

where m

ij

is a minor of A.

If A is an n � n matrix, then A

�1

an be found by numeri-

al methods, for example Gauss-Jordan elimination, Gaussian

elimination, and LU deomposition.


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Example 6. The Gaussian elimination proedure solves the

matrix equation Ax = b by �rst forming an augmented matrix

equation [Ab℄ and then transform this into an upper triangular

matrix

hn

a

0ij

o

b

0

i

, where a

0ij

are all zero exept when i � j.

Then,

x

i

=

1

a

0ii

0�

b

0i

�

k

X

j=i+1

a

0ij

x

j

1A


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The Gauss-Jordan elimination proedure �nds matrix inverse

by �rst forming a matrix [AI℄, and then use the Gaussian elim-

ination to transform this matrix into [I B℄. The result matrix

B is in fat A

�1

.


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The LU deomposition forms from the matrix A a produt LU

of two matries, one lower- while the other upper triangular.

This gives us three types of equation to solve, namely when

i < j, i = j and i > j, where i and j are the indies of row and

respetively olumn of the matrix produt. Then,

Ax = (LU)x = L(Ux) = b


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Letting y = Ux we have Ly = b, and therefore,

y

1

=

b

1

l

11

y

i

=

y

l

ii

0�

b

i

�

i�1

X

j=1

l

ij

y

j

1A

where i = 2; : : : ; n.


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Then sine Ux = y,

x

n

=

y

n

u

nn

x

i

=

1

n

ii

0�

y

i

�

n

X

j=i+1

u

ij

x

j

1A

where i = n� 1; : : : ; 1.


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Theorem 9. Let A and B be two square matries of equal

size. Then,

(AB)

�1

= B

�1

A

�1

Proof. Let C = AB. Then B = A

�1

C and A = CB

�1

,

therefore,

C = AB =

�

CB

�1

��

A

�1

C

�

= CB

�1

A

�1

C

Hene CB

�1

A

�1

= I, and thus B

�1

A

�1

= (AB)

�1

. {


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De�nition 14. The Einstein's summation is the simpli�a-

tion of notation by omitting a summation sign, keeping in mind

that repeated indies are impliitly summed over, for example

P

i

a

ik

a

ij

beomes

a

ik

a

ij

and

P

i

a

i

a

i

beomes

a

i

a

i


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De�nition 15. The multipliation of two matries A = fa

ij

g

and B = fb

ij

g is the matrix C = AB suh that

ik

= a

ij

b

jk


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Theorem 10. The matrix multipliation is assoiative.

Proof.

[(ab)℄

ij

= (ab)

ik

kj

= (a

il

b

lk

)

kj

= a

il

�

b

lk

kj

�

= a

il

(b)

lj

= [a(b)℄

ij

{


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Example 7. From Theorem 10, whih shows us the assoia-

tivity of matrix multipliation, we ould write the multiplia-

tion of three matries as [ab℄

ij

, whih is the same as writing

a

il

b

lk

kj

. And this applies in a similar manner to the multipli-

ation of four or more matries.


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Theorem 11. If A and B are two square and diagonal ma-

tries, then

AB = BA

But in general matrix multipliation is not ommutative.


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De�nition 16. A blok matrix is a matrix whih is is made

up of small matries put together, for example,

�

A B

C D

�

where A, B, C and D are matries.


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Theorem 12. Blok matries may be multiplied together in

the usual manner, for example,

�

A

1

B

1

C

1

D

1

� �

A

2

B

2

C

2

D

2

�

=

�

A

1

A

2

+B

1

C

2

A

1

B

2

+B

1

D

2

C

1

A

2

+D

1

C

2

C

1

B

2

+D

1

D

2

�

provided that all the produts involved are possible.


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De�nition 17. Let A = fa

ij

g be an n�n matrix. Then A is

alled a diagonal matrix if a

ij

= 0 when i 6= j. Here 1 � i; j �

n. In other words, a diagonal matrix has its omponents in the

form a

ij

=

i

Æ

ij

, where

i

is a onstant and Æ

ij

is the Kroneker

delta,

Æ =

�

0; i 6= j

1; i = j


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Theorem 13. A square matrix A an be diagonalised by the

transformation

A = PDP

�1

where P is made up of the eigenvetors of A and D is the

diagonal matrix desired.


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Example 8. Matrix diagonalisation an greatly help redu-

ing the number of parameters in a system of equations. For

instane, the systems Ax = y when diagonalised beomes

PDP

�1

x = y

that is Dx

0

= y

0

, where x

0

= P

�1

x and y

0

= P

�1

y. In this ase,

if A is an n � n matrix, we say that our new system obtained

through the proess of diagonalisation has anonialised from

n� n to n parameters.


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De�nition 18. A symmetri matrix is a square matrix A

whih satis�es

A

T

= A

Example 9. If A is a symmetri matrix, then

A

�1

A

T

= I


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De�nition 19. Let A be a square matrix. Then A is said to

be orthogonal if

AA

T

= I

Example 10. De�nition 19 is the same as saying that

A

�1

= A

T


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Theorem 14. A matrix A is symmetri if it an be expressed

as

A = QDQ

T

where Q is an orthogonal matrix and D is a diagonal matrix.


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Example 11. Any square matrix A may be deomposed into

two terms added together, that is A

s

+A

a

where A

s

is a symmet-

ri matrix and A

a

an antisymmetri matrix, alled respetively

a symmetri part and an antisymmetri part of A. Furthermore,

A

s

=

12

�

A+A

T

�

and,

A

a

=

12

�

A�A

T

�


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Deember, 2005

Documents

Kit · 12/10/2005 · C c b e three matrices. Then C = A + B is called the addition of matrices A and B if c ij = a + b for all i and j. Business mathematics, Matrix, 15 th b emer