I have a silly question: how to calculate the conditional number of a matrix? I should know it myself, but I graduated a few years ago and I have completly forgotten some basic facts.

Pawel

condition number of square matrix A = ||A|| ||inv(A)||

where ||.|| is some matrix norm and inv(A) is the matrix inverse of A. If you use one particular norm, which I dont recall at the moment, the condition number takes a simple form,

cond(A) = max |eigenvalue| / min |eigenvalue|

I think the norm you are refering to is the L2 norm. Condition numbers for matrices resulting from discretization of standard operators like the Laplacian can be easily estimated. Estimation of condition numbers for more general matrices requires numerical algorithms.

that's the one I'm familiar with. It's essentially the stiffness if the system.

The norm of a matrice is the product || A || * || A^-1 || where || .. || is a norm such as:

1) || A ||_1 = max(j) sum_i (a_ij)

2) || A ||_2 = sqrt( rho(A^t A ) ) where rho(A^t A ) is the spectral radius of A^t*A

3) || A ||_inf = max(i) sum_j( |a_ij| )

or when the matrix is symetric the condition number is given by:

max( | lambda_i |) / min ( | lambda_i | ) where lambda are the eigenvalues of A.

If I have Ax=b, when will I have to find the conditional number? and what is the physical meaning of conditional number and the spectral radius? Thank you very much.

Atit Koonsrisuk

In real life, You seldom (never) have to compute the condition number. For some systems, such as the discretization of a poisson type equation, the condition number can directly computed (or estimated) from known expressions.

The condition number estimates the effect of small perturbations on the B vector or the coefficients of the matrix A will have the solution. When the condition number is high, the small perturbations will cause big change on the solution. (this is the case for poisson type equations).

The spectral radius is the lagest eigenvalue of the matrix.