SOLUTION OF
SYSTEM OF LINEAR
EQUATIONS
Lecture 5: (a) ILL conditioned
equations.
(b) Infinity norm.
(c) Conditional number.
ILL Conditioned Equations
A system of equation for which small changes in the coefficients and/or constants
produce substantial changes in the solution is called ill-conditioned. Consider a
system of two linear equations in two unknowns:
 500 201 x   300 

   

 1000 401  y   300 
The solution of this system is x = 120, y = 300. We now make a small change in
the coefficient of a12 from 201 to 202. The solution of the system changes to
x = 39.8, y = 100, which shows that the solution is very sensitive to the value of
the coefficient matrix A. Thus, a small change in one of the element of the
coefficient A produce large change in the solution of the system of linear
equations, which are called ill-conditioned.
Infinity Norm
The infinity norm of  n  n  matrix is given by
A
max

i  1,2,3,..., n
n
a
ij
j 1
This norm is also called maximum row sum norm, since, by definition, it is the
maximum the sum of the absolute values of elements in each row. For example,
consider the matrix
 2 1 5 


A   3 0 9  , then
 8 4 1 


A
max

i  1,2,3,..., n
3
a
ij
j 1
3
Now,
a
1j
 a11  a12  a13  2  1  5  8
j 1
3
a
2j
 a21  a22  a23  3  0  9  12
j 1
3
a
3j
 a31  a32  a33  8  4  1  13
j 1
 A   max 8,12,13  13
Therefore, the infinity norm of the given matrix A is 13.
Conditional Number
The conditional number of an invertible square matrix A is given by
  A  cond  A  A  A1

However, if A is singular, then cond(A) = +. So, what this conditional number
do? It actually gives a measure of stability or sensitivity of a matrix to numerical
operations. It the conditional number of a matrix is near 1, it is said to be well
conditioned matrix and it is called ill-conditioned if its conditional number is much
higher than 1.
3
2 
1 
Let A   3 2.01 then A   100



1
Therefore,   A  cond  A   A  A
67

66.67 

100 
=5.01×200=1002,
which is much greater than 1. Thus, the matrix A is called ill-conditioned.