1. Matrix Condition Numbers
Scientific Computing
Matrix Condition Numbers

2. 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.

3. 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

4. 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

5. 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| )

6. 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.

7. 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

8. Condition Number And Accuracy
Class Practice: Show:

9. 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.