Definition of a Matrix

Aij

i = rowj = column

A

[ A ]

Definition of a Matrix

a11 a12 a13 … … a1n

a21 a22 a23 … … a2n

… … … aij … …

am1 am2 am3 … … amn

Definition of a Matrix

Size of a Matrix

a11 a12 a13 … … a1n

a21 a22 a23 … … a2n

… … … aij … …

am1 am2 am3 … … amn

size m x n

5 21 3 -740 -6 19 23-8 12 50 22

size 3 x 4

Size of a Matrix

Row Matrix

[ B ]m = 1

[ 50 -3 -27 35 ]

Column Matrix

-1033-615

{-10 33 -6 15}

{D}

n = 1

Square Matrix

a11 a12 a13 … … a1n

a21 a22 a23 … … a2n

… … … aij … …

an1 an2 an3 … … ann

size m x n5 21 3 40 -6 19 -8 12 50 size 3 x 3

m = n

Main Diagonal

a11 a12 a13 … … a1n

a21 a22 a23 … … a2n

… … … aij … …


5 21 3 40 -6 19 -8 12 50

5, -6, and 50 are diagonal elements

i = j

a11 a22 aij, …, …, ann

Symmetric Matrix

a11 a12 a13 … … a1n

a21 a22 a23 … … a2n

… … … aij … …


aij = aji

a12 = a21, a13 = a31, … a1n = an1

5 21 -3 21 6 19 -8 19 50

21, -3, and 19 are off-diagonal elements

Symmetric Matrix

Diagonal Matrixaij = 0, for a j

a11 0 0 … … 0

0 a22 0 … … 0

… … … aij … …

0 0 0 … … 0

a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

Diagonal Matrix

5 0 0 00 6 0 0 0 0 19 00 0 0 21

Unit or Identity Matrix

1 0 0 … … 00 1 0 … … 0 … … … aij … …

0 0 0 … … 1

aij = 1, for i = j aij = 0, for i j

a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

Unit or Identity Matrix

1 0 0 0 0 1 0 0 0 0 1 00 0 0 1

null matrix

aij =0

Matrix Operations

Equality

5 21 -3 A = 21 6 19

-8 19 50

5 21 -3 B = 21 6 19

-8 19 50

A = BAij = Bij

Addition and Subtraction

5 2 A = 2 6

-8 1

5 21 B = 21 6

-8 19

[A] + [B] = [C]

Aij + Bij = Cij

Addition and Subtraction

10 23 A+B = C = 23 12

-16 20

0 -19 A-B = C = -19 0

0 -18

Multiplication by ScalarScalar c, x [A]

5 2 A = 2 6

-8 1

15 6B = 6 18 -24 3

c = 3

c A = B

Multiplication of Matrices

-1 5 2 3 6A = B =

7 -3 4 –8 9

18 -43 51C =

2 45 -69

Conformable

[A] (m x n) x [B] (n x s) = [C] (m x s)

Aik x Bkj = C ij

Cij = Ai1B1j +ai2B2j+ … + AinBnj

Cij = AikBkj for k = 1 to n

Manual Multiplication

2 3 6 B =

4 –8 9

-1 5 18 -43 51A = C =

7 -3 2 45 -69

Application to Simultaneous Equations

a11x1 + a12x2 + a13x3 = P1

a12x2 + a22x2 + a23x3 = P2

a12x3 + a23x2 + a33x3 = P3

2x1 – 5x2 + 4x3 = 44

3x1 + 1x2 + -8x3 = -35

4x1 – 7x2 – 1x3 = 28

a11 a12 a13 x1 P1

a12 a22 a23 x2 = P2

a12 a23 a33 x3 P3



2 -5 4 x1 44

3 1 -8 x2 = -35

4 -7 -1 x3 -28

[A] {x} = {P}NOTES:[A] [B] [B] [A]A B C = (AB) C = A (BC)A (B + C) = AB + AC[A] [0] = [0], [0] [A] = [0]

Inverse of a Square Matrix

-2 1A-1 =

-1.5 0.5

Inverse of [A] = [A-1]

[A-1] [A] = [I][A] [A-1] = [I]

1 -2A = 3 4

Inverse of Square Matrix

1 0A A-1 =

0 1

Transpose of a Matrix

aijT = aji

a11 a21 a31 … … an1

a12 a22 a32 … … an2

… … … aji … …

a1n a2n a3n … … ann

Transpose of a Matrix

5 12 -3 18 21 6 19 16 -3 15 50 17 5 21 -3 12 6 15 -3 19 50 18 16 17

A (3 x 4) , AT (4 x 3)

Partitioning of Matrices

3 5 -1 ¦ 2 -2 4 7 ¦ 9 6 1 3 ¦ 4

1 8-5 2-3 6

7 -1

[A]

[B]


A11 ¦ A12

A = -----¦------- A21 ¦ A22

B = B11 ------

B21

A11 | A12 A11B11+A12B21

A= ---------------- AB= A21 | A22 A21B11+A22B21


B11

B = ------ B21

Partitioning of Matrices 19 28

A11B11 =

-43 34

14 -2A12B21 =

63 -9

A21B11 = [ -8 68 ]

A22B21 = [ 28 -4 ]

19 28 + 14 -2 -6 26

AB = -43 34 + 63 -9 = 20 25

[-8 68 ] + [28 -4] 20

64


A11B11+A12B21

AB = A21B11+A22B21

Solution of Simultaneous Equations by Gauss-Jordan Method

2x1 – 5x2 + 4x3 = 443x1 + x2 - 9x3 = -354x1 – 7x2 - x3 = 28

x1 – 2.5x2 + 2x3 = 223x1 + x2 - 8x3 = -354x1 - 7x2 - x3 = 28


x1 – 2.5x2 + 2x3 = 22 8.5x2 - 14x3 = -101

3x2 - 9x3 = -60

x1 – 2.5x2 + 2x3 = 22 x2 - 1.647x3 = -11.882

3x2 - 9x3 = -60


x1 – - 2.118x3 = -7.705 x2 - 1.647x3 = -11.882 - 4.059x3 = -24.354

x1 + 2.118x3 = - 7.705 x2 - 1.647x3 = -11.882

x3 = 6

x1 = 5 x2 = -2 x3 = 6


Check:

2(5) - 5(-2) + 4(6) = 443(5) +1(-2) - 8(6) = -354(5) - 7(-2) - 1(6) = 28

Matrix Inversion

[A] {x} = {C}

[A] [A] {x} = [A]-1 {C}

[A] [A] = [I]

{x} = [A] {C}

[A ¦ I ] { x ¦ -C }= 0

-1

-1

-1

[I ¦ B ] { x ¦ -C }= 0

{x} - [B] [C] = 0

{x} = [B] [C]

[B] = [A]

Matrix Inversion

-1

Method of Successive Transformations

2 4 3 ¦ 1 0 0 1 -2 0 ¦ 0 1 0-1 -4 5 ¦ 0 0 1

1 2 1.5 ¦ 0.5 0 0 1 -2 0 ¦ 0 1 0-1 -4 5 ¦ 0 0 1


1 2 1.5 ¦ 0.5 0 0 0 -4 -1.5 ¦ -0.5 1 0-1 -4 5 ¦ 0 0 1

1 2 1.5 ¦ 0.5 0 0 0 -4 -1.5 ¦ -0.5 1 0 0 -2 6.5 ¦ 0.5 0 1

1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0 0 0 7.25 ¦ 0.75 -0.5 1

1 2 1.5 ¦ 0.5 0 00 1 0.375 ¦ 0.125 -0.25 0

0 -2 6.5 ¦ 0.5 0 1


1 2 1.5 ¦ 0.5 0 0 0 1 0.375 ¦ 0.125 -0.25 0

0 0 1 ¦ 0.1034 -0.06897 0.1379


1 2 1.5 ¦ 0.5 0 00 1 0 ¦ 0.0862 -0.2241 -0.05170 0 1 ¦ 0.1034 -0.06897 0.1379

1 2 0 ¦ 0.3449 0.1034 -0.20690 1 0 ¦ 0.0862 -0.2241 -0.05170 0 1 ¦ 0.1034 -0.06897 0.1379


1 0 0 ¦ 0.1725 0.5516 -0.10350 1 0 ¦ 0.0862 -0.2241 -0.05170 0 1 ¦ 0.1034 -0.06897 0.1379

0.1725 0.5516 - 0.1035 A-1 = 0.0862 - 0.2241 - 0.0517 0.1034 - 0.06897 0.1379


l11 0 0 . . . . 0l21 l22 0 . . . . 0l31 l32 l33 0 . . . 0. . . . . . . .. . . . . . . .ln1 . . . . . . lnn

Cholesky Decomposition

Lower Triangular matrix [L]


[A] = [L] [L]T

[B] = [L]

[A] = ( [L] [L] )

[A] = [B] [B]

-1

-1-1

-1

T

T


Elements of [L]:

l = 0 for i<jl = (A - ∑l )l = (A - ∑l l )/l for i>j

Summation ∑ from r=1 to j-1

ij

ii

ij

ij

ij jr ir

ir 2 1/2


Elements of [B]:

b = 0 for i<jb = 1/lb = -(∑l l )/l or i>j

Summation ∑ from r=1 to i-1

ij

ii

ij

ii

ir ii rj


Example:

2 1 11 1.5 21 2 6.75


l = l = l = 0l = √2 = 1.414l = (1-0)/1.414 = 0.707l = (1-0)/1.414 = 0.707l = (1.5-0.707 ) = 1l = (2-0.7072)/1 = 1.5l = (6.75–(707 +1.5 )) ½ = 2.0

12

11

21

31

22

32

33

13 23

2 1/2

2 2


1.414 0.0 0.00.707 1.0 0.00.707 1.5 0.0

Cholesky Decompositionb = b = b = 0b = 1/1.414 = 0.707b = -(0.707 x 0.707)/1 = -0.5b = -(0.707x0.707+1.5(-0.5)/2 = 0.125b =1b = -(1.5 x 1)/2b = 0.5

[B]

12

11

21

31

22

32

33

13 23


0.707 0.0 0.0-0.5 1.0 0.00.125 -0.75 0.0

[A] = [B] [B]

[A] =

-1

-1

T


0.77 -0.594 0.063-0.5 1.563 -0.3750.063 -0.375 0.25

R = K rr = K R

R = {R R R R ….Rn}r = {r r r r ….rn}

-1

1 2 3 4

1 2 3 4

Definition of a Matrix

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