Cours De Matrices

CE 311 K - Introduction to Computer Methods

Matrices

Matrices  Variety of engineering problems lead to the 

need to solve systems of linear equations  Ax = b

 a11 a12   a21 a22 A =     am1 am 2

matrix

a1n   x1   b1       a2 n   x2   b2  x =   b =             am3  amn   xn   bm  a13 a23



column vectors

Matrix  Matrix

- a rectangular array of numbers arranged into m rows and n columns: Rows, i = 1, …, m

 a11 a12   a21 a22     am1 am 2

a1n   a23 a2 n      am3  amn  a13

Columns, j = 1, …, n



Examples  These

are valid matrices 1 0  0 1   

 These

a b  c e  

1 2 3  0 1 0   

are not

1 2 3   

 4  5 6  

 11   7    8 9  10   

Row and Column Matrices (vectors)  row

matrix (or row vector) is a matrix with one row r = ( r1 r2

 column

r3  rn )

vector is a matrix with only one

column  c1    c  c =  2     cm 

Square Matrix  When

the row and column dimensions of a matrix are equal (m = n) then the matrix is called square  a11 a12  a1n    a2 n   a21 a22 A =        an1 an 2  ann 

Transportation  Given

a list of cities (or destinations, nodes, etc) and flights (or roads, connections, etc) from city a to city b  Build a square matrix M with the cities indexing each side of the matrix – – –

M[a,b] = 1 if there is a connection from a to b M[b,a] = 1 if there is a reverse connection from b to a M[a,b] = 0 if there is no connection from a to b

Example 

– – – – – – – –

B to N, P, W, D A to N, W N to B, P, W, R, D, L P to N, B, W, R W to B, A, N, R, P, L R to N, P, W D to B, N L to N, W

Make a graph of this information where vertices represent cities and every edge represents a flight.



Albany (A), Boston (B), New York (N), Philly (P), Wash (W), Richmond (R), Detroit (D), and Las Vegas (L)

B 0 0 1 0 1 0 1 0

B A N P W R D L



A 0 0 0 0 1 0 0 0

N 1 1 0 1 1 1 1 1

P 1 0 1 0 1 1 0 0

W 1 1 1 1 0 1 0 1

R 0 0 1 1 1 0 0 0

D 1 0 1 0 0 0 0 0

Examine the matrix to see if there is a round trip between every city that is connected by a flight.

L 0 0 1 0 1 0 0 0

Matrix Transpose 

The transpose of the (m x n) matrix A is the (n x m) matrix formed by interchanging the rows and columns such that row i becomes column i of the transposed matrix  a11  a A =  21   a  m1

a12



a22 am 2

 

a1n   a2 n    amn  

 a11 a21  am1    a22 am 2  a AT =  12       a1n a2n  amn 

Example - Transpose 1 3 A=  2 5

1 3 4  A=  0 1 0 

1 2 T A =  3 5

1 0  AT = 3 1   4 0

Matrix Equality  Two

(m x n) matrices A and B are equal if and only if each of their elements are equal. That is A=B if and only if

aij = bij for i = 1,...,m; j = 1,...,n

Vector Addition  The

sum of two (m x 1) column vectors a and

b is  a1   b1   a1 + b1        a b a + b      2  a + b =  2 +  2 =  2            am   bm   am + bm 

Examples - Vector Addition 1 3 − 3 5 u=  v=  2  − 1 4 − 2      1   3   1 + 3   4 − 3  5  − 3 + 5 2 =  u+v =  +  =   2   − 1   2 − 1  1   4   − 2   4 − 2   2        

1  5  − 3 − 15  5u = 5  =   2   10   4   20     

Matrix Addition  a11   a21 A + B =    a  m1

a12 a22 am 2

 a11   a21 =   a  m1

 a1n   b11   a2 n   b21 +        b  amn   m1 + b11 a12 + b12 + b21 a22 + b22  + bm1 am 2 + bm 2

b12 b22 bm 2

 b1n   b2 n      bmn 

a1n + b1n   a2 n + b2 n       amn + bmn  

Examples - Matrix Addition 1 2 3  A = 2 1 4   1 4 3

 3 2 1 B = − 4 1 2    2 3 1 

1 2 3   3 2 1   4 4 4  A + B =  2 1 4 +  − 4 1 2  = − 2 2 6        1 4 3  2 3 1   3 7 4

 The

following matrix addition is not defined 1 2 2 4 6 5 2 + 1 3 5 = ? (not defined!)    

Scalar – Matrix Multiplication  Multiplication

defined as

 Examples

of a matrix A by a scalar is  αa11 αa12  αa1n    α a α a α a  22 2n  αA =  21       αam1 αam 2  αamn 

α = 4,

1 2 4 8  A=  , αA =  0 4  0 1    

1 4 1  − 2 − 8 − 2 α = −2, A =   , αA =  − 4 0 − 6  2 0 3    

Matrix – Matrix Multiplication 

The product of two matrices A and B is defined only if –



the number of columns of A is equal to the number of rows of B.

If A is (m x p) and B is (p x n), the product is an (m x n) matrix C C mxn = Amxp B pxn

Matrix – Matrix Multiplication  a11 a12  a1 p  b11 b12  b1n     a1 p  b21 b22 b1n   a21 a22 C = AB =              am1 am 2  amp  b p1 b p 2  b pn      a11b11 +  + a1 p b p1 a11b12 +  + a1 p b p 2  a11b1n +  + a1 p b pn    a21b1n +  + a2 p b pn   a21b11 +  + a2 p b p1 a21b12 +  + a2 p b p 2 =         am1b11 +  + amp b p1 am1b12 +  + amp b p 2  am1b1n +  + amp b pn    p

cij = ∑ aik bkj k =1

Example - Matrix Multiplication

a = (a1, a2 ,, an )

 b1     b2  b=      bn 

c1x1 = a1xn bnx1 (scalar)

 b1     b2  c = a ⋅ b = (a1, a2 ,, an )  = a1b1 + a2b2 +  + an bn     bn 

Example - Matrix Multiplication  a11 a12  a1n  a a22  a2n  21  A=    a   n1 an 2  ann 

 b1     b2  b=      bn 

c nx1 = Anxn bnx1

 a11 a12  a1n  b1  a11b1 + a12b2 +  a1n bn   a  21 a22  a2n   b2  a11b1 + a22b2 +  a2n bn  c = Ab = =         a   n1 an 2  ann  bn  an1b1 + an 2b2 +  ann bn

Example - Matrix Multiplication

1 3  A=  2 4

 2 1 B=  3 5

C 2 x 2 = A2 x 2 B2 x 2

1 3 2 1  1 ⋅ 2 + 3 ⋅ 3 1 ⋅1 + 3 ⋅ 5  11 16  C = A⋅ B =  = =      2 4 3 5 2 ⋅ 2 + 4 ⋅ 3 2 ⋅1 + 4 ⋅ 5 16 22

Example - Matrix Multiplication 1 2 3  A =  2 1 4   1 4 3

2 1  B = 1 2    2 1 

1 2 3   2 1  C 3 x 2 = A3 x3 B3 x 2 C = A ⋅ B =  2 1 4 1 2    1 4 3 2 1  1 ⋅ 2 + 2 ⋅1 + 3 ⋅ 2 1 ⋅1 + 2 ⋅ 2 + 3 ⋅1 10 8  = 2 ⋅ 2 + 1 ⋅1 + 4 ⋅ 2 2 ⋅1 + 1 ⋅ 2 + 4 ⋅1 = 13 8      1 ⋅ 2 + 4 ⋅1 + 3 ⋅ 2 1 ⋅1 + 4 ⋅ 2 + 3 ⋅1 12 12

Example - Matrix Multiplication 3 0  A = 1 1    5 2

4 7  B=  6 8  C 3 x 2 = A3 x 2 B2 x 2

3 0  12 21 4 7    C = A⋅ B = 1 1  = 10 15    6 8    5 2 32 51

Diagonal Matrices





Diagonal Matrix

Tridiagonal Matrix

0  a11 0   0 a22 0 A =  0 0   0 0  0  a11 a12   a21 a22 A =  0 a32  0  0  0 0 

0 a23  0

0   0  0   ann  0 0   a54

0   0  0   a45  a55 

Identity Matrix 



Identity Matrix 1 0 I = 0 0 

0

0

1 0

0 1

0

0

0 0  0 1 

The identity matrix has the property that if A is a square matrix, then IA = AI = A

Matrix Inverse  If

A is an (n x n) square matrix and there is a matrix X with the property that AX = I

X

is defined to be the inverse of A and is denoted A-1 AA−1 = I

A−1 A = I

Example - Matrix Inverse

AA −1 = I

• Example (2 x 2) matrix  a11 a12  A =  a a  21 22 

A−1 =

 a22 − a12  1 a11a22 − a12 a21 − a21 a11 

Special Matrices



Upper Triangular Matrix



Lower Trangular Matrix

u11 u12    0 u22 U= 0  0  0 0 

u13 u14  u23 u24   u33 u34  0 u44 

0  L11    L21 L22 L=  L31 L32 L  41 L42

0 0 L33 L43

0  0   0  L44 