Matrix norm induced by an energy norm ($\Vert B{\Vert }_A=\Vert A^{1/2}B{A}^{-1/2}{\Vert }_2$)

For a real symmetric positive-definite matrix $A$, the energy norm is $\Vert x{\Vert }_A=(x^{T}Ax{)}^{1/2}=\Vert A^{1/2}x{\Vert }_2$. 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.