represent respectively the i-th row and the j-th column of the matrix (1), and aij
represents the ... We now consider the question of arithmetic involving matrices.
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(1959): Vision screening for elementary schools: the orida study .Berekely,CA,University of. California . .... Jaeb Center for Health Research, Tampa FL.
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1. Chapter Four: Matrices Theory. References: 1. Advanced Engineering
Mathematics by C. Ray Wylie. 2. Advanced Engineering Mathematics by Erwin
Kreyszig.
University of Technology Dep. Of Electrical & Electronic Eng. www.uotiq.org Lecture (1)
Engineering Analysis Lecture notes Third year Lec. Dr. Abbas H. Issa
Chapter Four: Matrices Theory References: 1. Advanced Engineering Mathematics by C. Ray Wylie 2. Advanced Engineering Mathematics by Erwin Kreyszig
4.1 Definition: A matrix of order (m x n), or m by n is a rectangular matrix, array of numbers having m rows and n columns. It can be written in the form
⎡ a11 ⎢a A = ⎢ 21 ⎢ M ⎢ ⎣ a m1
a12
L
a 22
L L L
a1 n ⎤ a 2 n ⎥⎥ M ⎥ ⎥ a mn ⎦
A ∈ R m×n
If m = n it is called square matrix of order m or n. 4.2 Diagonal matrix: a square matrix D is said to be diagonal matrix if the element of the matrix satisfy
d
ij
= 0
(i ≠
j)
d
ij
≠ 0
(i =
j)
⎡2 D = ⎢⎢ 0 ⎢⎣ 0
0 − 1 0
0 ⎤ 0 ⎥⎥ 5 ⎥⎦
A matrix L is called a lower triangular matrix of order 3
⎡2 L = ⎢⎢ 3 ⎢⎣ 5
0 −1 4
0⎤ 0 ⎥⎥ 3 ⎥⎦
A matrix U is called an upper triangular matrix of order 3
University of Technology Dep. Of Electrical & Electronic Eng. www.uotiq.org Lecture (1)
Engineering Analysis Lecture notes Third year Lec. Dr. Abbas H. Issa
4.3 Equality of matrices: Two matrices A= (ajk) and B= (bjk) of the same order are equal iff.
ajk= bjk 4.4 Addition and subtraction of matrices: To add or subtract, two matrices must be of the same order if
[ ]
A = a ij
[
[ ]
, B = bij
A m B = a ij m bij
]
then
Note that A+B = B+A
(Commutative law)
(A+B) +C = A+ (B+C)
(Associative law)
4.5 Multiplication of matrix by a scalar: If A= [aij] and q is a scalar number, then qa = q [aij] 4.6 Multiplication of matrices: let
[ ]
A = a jk ∈ R m×n
and
[ ]
B = b jk ∈ R r × p
Then (AB) is defined only when n = r and it is (m x p) matrix
[ ]
C = c jk
,
c jk =
n
∑ (a i =1
ji
b ik )
>>> Properties of matrix operation: ( kA) B = k ( AB ) (associativ e A ( BC ) = ( AB ) C = ABC and distributi ve) ( A + B ) C = AC + BC C ( A + B ) = CA + CB AB ≠ BA ( Not commulativ e)
A . B → B is pre-multiplied by A or A is post- multiplied by B _____________________________________________________________________ 2
University of Technology Dep. Of Electrical & Electronic Eng. www.uotiq.org Lecture (1)
Engineering Analysis Lecture notes Third year Lec. Dr. Abbas H. Issa
4.7 Transpose of a matrix: If A = a jk
then
AT = akj
Note that ( A + B )T = AT + B T ( A B )T = B T AT
and
( AT )T = A
4.8 Symmetric and skew – symmetric matrices: If AT=A → symmetric matrix (A is square matrix) If AT=-A → skew- symmetric matrix e.g. ⎡ 1 − 3⎤ B=⎢ ⎥ is symmetric ⎣− 3 2 ⎦
⎡0 − 1⎤ A=⎢ ⎥ is skew − symmetric ⎣1 0 ⎦
4.9 Principle diagonal & trace: If A= [ajk] is a square matrix, then the diagonal which contains all elements of ajk (j=k) is called the principle or main diagonal. The sum of these elements is called the trace e.g. 0 − 1⎤ ⎡1 ⎢ A = ⎢ 2 − 3 5 ⎥⎥ , The trace of A = 1 + ( −3) + 6 = 4 ⎢⎣− 4 1 6 ⎥⎦
4.10 Unity matrix: It is a square matrix that all elements of its principle diagonal are 1 while the other is 0 AI = I A = A
4.11 Determinant: If A is a square matrix the ∆A=|A| is the determinant of A. found as follows: a- Minor given any element of ajk of ∆ we associate a new determinant of order (n-1) obtained by removing all elements of the JM row and KM column. This is called the Minor of ajk
University of Technology Dep. Of Electrical & Electronic Eng. www.uotiq.org Lecture (1)
Engineering Analysis Lecture notes Third year Lec. Dr. Abbas H. Issa
Example: 2 −1 1 −3 2 5 1 0 −2 4 2 3
3 0 2 1 2 −1 3
The minor of element 5 is 1
0
2 =Min 23
4
2
1
b- Cofactor: if we multiply the minor of ajk by (-1)j+k the result is called the cofactor of ajk and it is denoted by Ajk e.g. the cofactor of element 5 in last example is ( −1)
University of Technology Dep. Of Electrical & Electronic Eng. www.uotiq.org Lecture (1)
Engineering Analysis Lecture notes Third year Lec. Dr. Abbas H. Issa
And if we take the transpose of C then CT is called the adjoint of the original matrix A (Adj A). Example: ⎡2 3 5⎤ A = ⎢⎢4 1 6⎥⎥ , then ⎣⎢1 4 0⎥⎦ ⎡ 1 ⎢+ ⎢ 4 ⎢ 3 C = ⎢− ⎢ 4 ⎢ 3 ⎢+ 1 ⎣
⎡− 24 20 13 ⎤ Adj A = C T = ⎢⎢ 6 − 5 8 ⎥⎥ ⎢⎣ 15 − 5 − 10⎥⎦
4.13 Matrix Inversion: If A is a non-singular matrix of order n (i.e ∆A≠0), then there exists a unique inverse A-1 such that AA-1=1 and can be expressed as Example: from previous example det A=45 ⎡ − 24 ⎢ 45 ⎢ 6 A −1 = ⎢ ⎢ 45 ⎢ 15 ⎢⎣ 45
20 45 −5 45 −5 45
13 ⎤ 45 ⎥ 8 ⎥ ⎥ 45 ⎥ − 10 ⎥ 45 ⎥⎦
Note that: 1. The inverse of 2x2 matrix ⎡ a11 a12 ⎤ A=⎢ ⎥ ⎣a 21 a 22⎦ 1 ⎡ a 22 − a12⎤ is A −1 = det A ⎢⎣− a 21 a11 ⎥⎦
2. The inverse of a non-singular diagonal matrix is
4.14 Rank: The rank of a matrix A is the largest value of r for which there exists an (r x r) sub matrix of A with non-vanishing determinant. Example: The matrix ⎡ 1 2 −1 3 ⎤ ⎢ 3 4 0 − 1⎥ ⎥ ⎢ ⎢− 1 0 − 2 7 ⎥ ⎥ ⎢ ⎦ ⎣
, is of the rank 2, since each of the third-order sub matrices ⎡2 − 1 3 ⎤ ⎢4 0 − 1⎥, ⎥ ⎢ ⎢⎣0 − 2 7 ⎥⎦
⎡ 1 −1 3 ⎤ ⎢3 0 − 1⎥⎥, ⎢ ⎢⎣− 1 − 2 7 ⎥⎦
⎡1 2 3⎤ ⎢ 3 4 − 1⎥, ⎥ ⎢ ⎢⎣− 1 0 7 ⎥⎦
⎡ 1 2 − 1⎤ ⎢3 4 0 ⎥ ⎥ ⎢ ⎢⎣− 1 0 − 2⎥⎦
, is singular, while not all second-order sub matrices are singular. 4.15 Elementary operation: (Gaussian elimination method) 1. Interchanging columns or rows 2. Multiplication of row (column) by a non-zero number. 3. Addition to (or subtraction from) all the elements of any row (column) k times the corresponding elements of any other row (column). We will use elementary row operation only ERO:
