Scientific Computing
Matrix Condition Numbers
Matrix Condition Number
• Multiplication of a vector x by a matrix A
results in a new vector Ax that can have a very
different norm from x.
• The range of the possible change can be
expressed by two numbers,
•
=||A||
• Here the max, min are over all non-zero vectors x.
Matrix Condition Number
• Definition: The condition number of a
nonsingular matrix A is given by:
κ(A) = M/m
by convention if A is singular (m=0) then κ(A) = ∞.
• Note: If we let Ax = y, then x = A-1 y and
m  min
Ax
x
 min
y
1
1

1
A y
A y
max
y

1
A1
Matrix Condition Number
• Theorem: The condition number of a
nonsingular matrix A can also be given as:
κ(A) = || A || * || A-1||
• Proof: κ(A) = M/m. Also, M = ||A|| and by
the previous slide m = 1 / (||A-1 ||). QED
Properties of the Matrix Condition
Number
•
•
•
•
For any matrix A, κ(A) ≥ 1.
For the identity matrix, κ(I) = 1.
For any permutation matrix P, κ(P) =1.
For any matrix A and nonzero scalar c ,
κ(c A) = κ(A).
• For any diagonal matrix D = diag(di),
κ(D) = (max|di|)/( min | di| )
What does the condition
number tell us?
• The condition number is a good indicator of
how close is a matrix to be singular. The
larger the condition number the closer we
are to singularity.
• It is also very useful in assessing the accuracy
of solutions to linear systems.
• In practice we don’t really calculate the
condition number, it is merely estimated , to
perhaps within an order of magnitude.
Condition Number And Accuracy
• Consider the problem of solving Ax = b. Suppose b
has some error, say b + δb. Then, when we solve the
equation, we will not get x but instead some value
near x, say x + δx.
A(x + δx) = b + δb
• Then, A(x + δx) = b + δb
Condition Number And Accuracy
• Class Practice: Show:
Condition Number And Accuracy
• The quantity ||δb||/||b|| is the relative
change in the right-hand side, and the
quantity ||δx||/||x|| is the relative error
caused by this change.
• This shows that the condition number is a
relative error magnification factor. That is,
changes in the right-hand side of Ax=b can
cause changes κ(A) times as large in the
solution.