Editing Matrix norm (section)

===Energy norms===

If the vector norms <math>\|\cdot\|_{\alpha}</math> and <math>\|\cdot\|_{\beta}</math> are given in terms of [[Norm_(mathematics)#Energy_norm|energy norms]] based on [[Symmetric_matrix|symmetric]] [[Definite_matrix|positive definite]] matrices <math>P</math> and <math>Q</math> respectively, the resulting operator norm is given as
<math display="block">
\|A\|_{P, Q} = \sup \{ \|Ax\|_Q : \|x\|_P \leq 1 \}.
</math>

Using the symmetric [[Square_root_of_a_matrix|matrix square roots]] of <math>P</math> and <math>Q</math> respectively, the operator norm can be expressed as the spectral norm of a modified matrix:

<math display="block">
\|A\|_{P, Q} = \|Q^{1/2} A P^{-1/2}\|_{2}.
</math>