Matrix norms#
Frobenius norm#
Let \(\boldsymbol A \in \mathbb R^{m\times n}\). Frobenius norm of \(\boldsymbol A\) is defined as
Spectral norm#
The spectral norm of the matrix \(\boldsymbol A \in \mathbb R^{m\times n}\) is equal to maximal eigenvalue of \(\boldsymbol A^\top \boldsymbol A\):
Condition number#
For a square matrix \(\boldsymbol A\) define its conditional number as
If \(\boldsymbol A\) is singular then \(\kappa(\boldsymbol A) = \infty\).
Properties of conditional numbers
\(\kappa(\boldsymbol A) \geqslant 1\);
\(\kappa(\boldsymbol A) = \kappa(\boldsymbol A^{-1})\);
\(\kappa(\boldsymbol{AB}) \leqslant \kappa(\boldsymbol A)\kappa(\boldsymbol B)\);
\(\kappa(\boldsymbol A) = \Big\vert\frac{\lambda_{\max}(\boldsymbol A)}{\lambda_{\min}(\boldsymbol A)}\Big\vert\) if \(\boldsymbol A^\top = \boldsymbol A\).