A ij i = row j = column A [ A ] Definition of a Matrix

a 11 a 12 a 13 … … a 1n a 21 a 22 a 23 … … a 2n … … … a ij … … a m1 a m2 a m3 … … a mn Definition of a Matrix

Size of a Matrix a 11 a 12 a 13 … … a 1n a 21 a 22 a 23 … … a 2n … … … a ij … … a m1 a m2 a m3 … … a mn size m x n

size 3 x 4 Size of a Matrix

Row Matrix [ B ] m = 1 [ ]

Column Matrix { } {D} n = 1

Square Matrix a 11 a 12 a 13 … … a 1n a 21 a 22 a 23 … … a 2n … … … a ij … … a n1 a n2 a n3 … … a nn size m x n size 3 x 3 m = n

Main Diagonal a 11 a 12 a 13 ……a 1n a 21 a 22 a 23 ……a 2n ………a ij …… a n1 a n2 a n3 ……a nn , -6, and 50 are diagonal elements i = j a11a22aij, …, …, ann

Symmetric Matrix a 11 a 12 a 13 ……a 1n a 21 a 22 a 23 ……a 2n ………a ij …… a n1 a n2 a n3 ……a nn a ij = a ji a12 = a21, a13 = a31, … a1n = an1

, -3, and 19 are off-diagonal elements Symmetric Matrix

Diagonal Matrix a ij = 0, for a  j a 11 00……0 0a 22 0……0 ………a ij …… 000……0 a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

Diagonal Matrix

Unit or Identity Matrix 100……0 010……0 ………a ij …… 000… …1 a ij = 1, for i = j a ij = 0, for i  j a12 = a21 = 0, a13 = a31 = 0, … a1n = an1= 0

Unit or Identity Matrix null matrix a ij =0

Matrix Operations

Equality A = B = A = B A ij = B ij

Addition and Subtraction 5 2 A = B = [A] + [B] = [C] A ij + B ij = C ij

Addition and Subtraction A+B = C = A-B = C =

Multiplication by Scalar Scalar c, x [A] 5 2 A = B = c = 3 c A = B

Multiplication of Matrices A = B = – C = Conformable [A] (m x n) x [B] (n x s) = [C] (m x s) A ik x B kj = C ij C ij = A i1 B 1j +a i2 B 2j + … + A in B nj Cij =  A ik B kj for k = 1 to n

Manual Multiplication B = 4 – A = C =

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

a 11 a 12 a 13 x 1 P 1 a 12 a 22 a 23 x 2 = P 2 a 12 a 23 a 33 x 3 P 3 Application to Simultaneous Equations

x x 2 = x [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 1 A -1 = Inverse of [A] = [A -1 ] [A -1 ] [A] = [I] [A] [A -1 ] = [I] 1 -2 A = 3 4

Inverse of Square Matrix 1 0 A A-1 = 0 1

Transpose of a Matrix a ij T = a ji a 11 a 21 a 31 ……a n1 a 12 a 22 a 32 ……a n2 ………a ji …… a 1n a 2n a 3n ……a nn

Transpose of a Matrix A (3 x 4), A T (4 x 3 )

Partitioning of Matrices ¦ ¦ ¦ [A] [B]

Partitioning of Matrices A 11 ¦ A 12 A= -----¦ A 21 ¦ A 22 B= B B 21

A 11 | A 12 A 11 B 11 +A 12 B 21 A= AB= A 21 | A 22 A 21 B 11 +A 22 B 21 Partitioning of Matrices B 11 B = B 21

Partitioning of Matrices A 11 B 11 = A 12 B 21 = A21B11 = [ ] A22B21 = [ ]

AB = = [-8 68 ] + [28 -4] Partitioning of Matrices A 11 B 11 +A 12 B 21 AB = A 21 B 11 +A 22 B 21

Solution of Simultaneous Equations by Gauss-Jordan Method 2x1 – 5x2 + 4x3 = 44 3x1 + x2 - 9x3 = -35 4x1 – 7x2 - x3 = 28 x1 – 2.5x2 + 2x3 = 22 3x1 + x2 - 8x3 = -35 4x1 - 7x2 - x3 = 28

Solution of Simultaneous Equations by Gauss-Jordan Method x1 – 2.5x2 + 2x3 = x2 - 14x3 = x2 - 9x3 = -60 x1 – 2.5x2 + 2x3 = 22 x x3 = x2 - 9x3 = -60

Solution of Simultaneous Equations by Gauss-Jordan Method x1 – x3 = x x3 = x3 = x x3 = x x3 = x3 = 6 x1 = 5 x2 = -2 x3 = 6

Solution of Simultaneous Equations by Gauss-Jordan Method Check: 2(5) - 5(-2) + 4(6) = 44 3(5) +1(-2) - 8(6) = -35 4(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

[I ¦ B ] { x ¦ -C }= 0 {x} - [B] [C] = 0 {x} = [B] [C] [B] = [A] Matrix Inversion

Method of Successive Transformations ¦ ¦ ¦ ¦ ¦ ¦ 0 0 1

Method of Successive Transformations ¦ ¦ ¦ ¦ ¦ ¦

¦ ¦ ¦ ¦ ¦ ¦ Method of Successive Transformations

¦ ¦ ¦ Method of Successive Transformations ¦ ¦ ¦

1 2 0 ¦ ¦ ¦ Method of Successive Transformations ¦ ¦ ¦

A-1 = Method of Successive Transformations

l l21 l l31 l32 l ln lnn Cholesky Decomposition Lower Triangular matrix [L]

Cholesky Decomposition [A] = [L] [L]T [B] = [L] [A] = ( [L] [L] ) [A] = [B] [B] T T

Cholesky Decomposition Elements of [L]: l = 0 for i<j l = (A - ∑l ) l = (A - ∑l l )/l for i>j Summation ∑ from r=1 to j-1 ij ii ij jrir 21/2

Cholesky Decomposition Elements of [B]: b = 0 for i<j b = 1/l b = -(∑l l )/l or i>j Summation ∑ from r=1 to i-1 ij ii ij ii ir ii rj

Cholesky Decomposition Example:

Cholesky Decomposition l = l = l = 0 l = √2 = l = (1-0)/1.414 = l = ( ) = 1 l = ( )/1 = 1.5 l = (6.75–( )) ½ = /2 2 2

Cholesky Decomposition

Cholesky Decomposition b = b = b = 0 b = 1/1.414 = b = -(0.707 x 0.707)/1 = -0.5 b = -(0.707x (-0.5)/2 = b =1 b = -(1.5 x 1)/2 b = 0.5 [B]

Cholesky Decomposition [A] = [B] [B] [A] = T

Cholesky Decomposition R = K r r = K R R = {R R R R ….Rn} r = {r r r r ….rn}