Contents

MATRIX ALGEBRA


1. Vectors and Matrices

Example 1: Use of matrices in solving linear systems of equations

1.1. Definitions and terminology

The word 'number' means a real or a complex number.
R = {real numbers}, C = {complex numbers}
Rn = {(x1 , x2 , ..., xn ) | x1 , x2 , ..., xn R}
Cn = {(x1 , x2 , ..., xn ) | x1 , x2 , ..., xn C}
A vector = (x1 , x2 , ..., xn ) is often written as

= (x1 x2 ... xn )

or

= x1
x2
:
xn

These two are special cases of a matrix

A = a11 a12 ···a1n .
a21 a22 ···a2n
: : :
am1 am2 ···amn

The numbers aij are called the entries or the elements of the matrix A.

Note that in the double-subscript notation for the elements, the subscript i denotes the row while j denotes the column in which the element aij stands.
The dimension or the size of the above matrix A is m x n. One often writes Am x n or

A = (aij ) = (aij )m x n .

If m = n then A is a square matrix.
We also say that a square matrix A is

1) an upper triangular matrix, if aij = 0 when i > j.

x x x x x
0 x x x x
0 0 x x x
0 0 0 x x
0 0 0 0 x

2) a lower triangular matrix, if aij = 0 when i < j.

x 0 0 0 0
x x 0 0 0
x x x 0 0
x x x x 0
x x x x x

3) a diagonal matrix, if aij = 0 when i j.

x 0 0 0 0
0 x 0 0 0
0 0 x 0 0
0 0 0 x 0
0 0 0 0 x

Only the elements on the main diagonal can be 0.

4) A tridiagonal matrix, if aij = 0 when | i - j | > 1.

x x 0 0 0
x x x 0 0
0 x x x 0
0 0 x x x
0 0 0 x x

5) An upper Hessenberg matrix, if aij = 0 when i > j + 1.
A lower Hessenberg matrix, if aij = 0 when j > i + 1.

6) A Toeplitz matrix if aij = a( i - j ), for example

a(0) a(-1) a(-2) a(-3)
a(1) a(0) a(-1) a(-2)
a(2) a(1) a(0) a(-1)
a(3) a(2) a(1) a(0)

A diagonal matrix is often written as

1 0 0 0 0 = diag (1 , 2 , 3 , 4 , 5 ).
0 2 0 0 0
0 0 3 0 0
0 0 0 4 0
0 0 0 0 5

The identity matrix of dimension n, denoted by In (or I, if the dimension is clear from the context), is

In = diag (1, 1, ..., 1) = 1 0 0 0 ··· 0
0 1 0 0 ··· 0
0 0 1 0 ··· 0
:
0 0 ··· 0 1

I = (ij ) , ij = 1, kun i = j
0, kun i j

A zero matrix O (or Om x n ) is a matrix whose elements are all zeros.


Exercises: E6
Contents
Next section