# How to find the condition number of a 3*3 matrix with singular value 10, 1 and 1/10? [closed]

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?

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).