= Matrix norm induced by an energy norm
{title2=$\|B\|_A=\|A^{1/2}BA^{-1/2}\|_2$}
For a real <symmetric positive-definite matrix> $A$, the <energy norm> is $\|x\|_A=(x^TAx)^{1/2}=\|A^{1/2}x\|_2$. In the defining <supremum> of the <induced matrix norm>, substitute $y=A^{1/2}x$. The <matrix square root> is an <invertible matrix>, so this ranges over every nonzero <vector> and proves the displayed identity. This <matrix similarity> is useful because contraction in the <energy norm> becomes an ordinary Euclidean <operator norm> estimate after a change of coordinates.
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