0
$\begingroup$

If the singular values of a 3*3 matrix are 10, 1 and 1/10 respectively, what is the condition number of the matrix? Is such a matrix numerically tractable?

$\endgroup$
1
$\begingroup$

the condition number of a matrix, $A$, with respect to a given matrix norm is: $\kappa(A) = ||A||\cdot||A^{-1}||$. in this case the singular values are the square root of the non-negative eigenvalues of $A^*A$, and the largest of them is the operator norm of $A$, so under this norm $||A|| = 10$ and $||A^{-1}|| = \left(\frac{1}{10}\right)^{-1} = 10$. Thus the condition number, $\kappa(A) = ||A||\cdot||A^{-1}|| = 10\cdot 10 = 100$. In general smaller condition numbers mean matrices are more tractable (Identity has condition number 1). A condition number this high suggests conditioning is needed (SOR, Gauss-Sidel, etc).

| cite | improve this answer | |
$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.