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Chapter 1
Matrices
1.1 Basic Properties of Matrices
A matrix A is a rectangular array of elements denoted by
 a11
a
21
A= 
 

 a m1
a12
a 22

am 2
a1n 
... a 2 n 


 

... a mn 
...
(1.1-1)
This matrix A is called an mn matrix or a matrix of order “m by n” referring to the fact
that A has m rows and n columns. For a square matrix m is equal to n. Element aij is located in
the ith row and jth column.
(1) Matrix equality: A = B if they are of the same order and aij = bij
(2) Column matrix [m, 1] has m rows and one column.
 1 
A[3, 1] =  3 
 
  2
(3) Row matrix [1, n] has 1 rows and n column.
A[3, 1] = 1 3  2
(4) Transposition: When a matrix is transposed, the rows and column are interchanged.
1 2 3
T
A= 
 A =
4
5
6


1 4 
2 5 ; (a )T = a
ij
ji


 3 6
(5) Symmetric matrix: symmetric matrix is one that is equal to its own transpose.
 1  1 2
A = A or aij = aji. Example: A =  1 5 4


4 6
 2
T
1
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 1  1 2
(6a) Skew matrix: aij =  aji; and not all aii = 0. Example: A =  1
5 4


 2  4 6
 0  1 2
(6b) Skew-symmetric matrix: aij =  aji; and aii = 0. Example: A =  1
0 4


 2  4 0
(7) Complex conjugate of a matrix A is a matrix derived from A by taking the complex
conjugate of all the elements in A.
2 
3  i 2 
3  i
A= 
 A = 


  2i 1  i 
 2i 1  i 
(8) Hermitian matrix is equal to the transpose of its own complex conjugate.
3  i
3  i
 4
 4
A= 
 A = 

2 
2 
3  i
3  i
3  i
 4
T
A* = A  = 
=A
2 
3  i
(9) Skew-Hermitian matrix is equal to negative of the transpose of its own complex
conjugate.
 2  i
 2  i
2  i
 0
 0
 0
T
A= 

=

=
A


A
2  i
 2  i
0 
0 
0 
2  i


 2  i
 0
T
A =  A  = 
0 
2  i
(10) Zero or null matrix has all its elements zero and is denoted by 0, or, aij = 0 for all i
and j.
(11) Identity or unit matrix I has the following properties
aij = ij = 1 if i = j
aij = ij = 0 if i  j
(12) Matrix addition and subtraction is defined only for matrices of the same order.
C = A + B, cij = aij + bij
2
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D = A  B, dij = aij  bij
(13) Matrix multiplication by a scalar
C = kA  cij = kaij
(14) Matrix multiplication
A[m, p] B[p, n] = C[m, n]
A is said to be “post-multiplied” by B, or B is “pre-multiplied” by A, to distinguish the
order in which A and B appear. The element cij is obtained from the multiplication of the row i of
matrix A and the column j of matrix B.
p
cij =
a
k 1
b
ik kj
1 2 3
Example: A = 
, B =
4 5 6
1  1 1  1
2 0 3 1 


0 1 2 3 
A[2, 3] B[3, 4] = C[2, 4]
3
c11 =
a
k 1
b = a11b11 + a12b21 + a13b31 = (1)(1) + (2)(2) + (3)(0) = 5
1k k 1
3
c23 =
a
k 1
Similarly,
b = a21b13 + a22b23 + a23b33 = (4)(1) + (5)(3) + (6)(2) = 31
2k k 3
 5 2 13 10
C= 

14 2 31 19
(15) Matrix partition
 a11
a
21
A= 
a 31

a 41
a12
a13
a14
a 22
a 23
a 24
a 32
a 33
a 34
a 42
a 43
a 44
 a11 a12
a
a 22
If A11 =  21
a31 a32


a13
a 23
a33
a15 
a 25 

a 35 

a 45 


 , A12 =



3






a14
a 24
a34
a15 
a 25 

a35 


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

A21 = 


a 41
A
Then A =  11
 A21
a 42
a 43


 , A22 =



  A11 T
A12 
T
and A = 
T
A22 
 A12 






a 44





a 45 
 A21 T 
 A22 T 
The following example shows the use of Matlab commands for matrix partition.
Matlab Example __________________________________________________
Example 1.1
>> A=[1 2 3 4 5;2 -1 2 5 6;3 2 -3 4 1;4 -1 -2 1 1]
A=
1 2 3 4 5
2 -1 2 5 6
3 2 -3 4 1
4 -1 -2 1 1
>> A11=A(1:3,1:3)
A11 =
1 2 3
2 -1 2
3 2 -3
>> A12=A(1:3,4:5)
A12 =
4 5
5 6
4 1
5
>> A21=A(4,1:3)
A21 =
4 -1 -2
>> A22=A(4,4:5)
A22 =
1 1
>> At=A'
At =
1 2
2 -1
3 2
4 5
5 6
3
2
-3
4
1
4
-1
-2
1
1
4
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>> At=[A11' A21';A12' A22']
At =
1 2 3 4
2 -1 2 -1
3 2 -3 -2
4 5 4 1
5 6 1 1
A set of n simultaneous linear equations with n unknowns can be written in matrix form as
Ax = b
where A is the coefficient matrix, x is the column vector of the unknowns, and b is the column
vector of the right hand side. For a set of two equations and two unknowns
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
a12 
a
A =  11
, x =
a
a
21
22


 x1 
 x  , and b =
 2
 b1 
b 
 2
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