Induced matrix norm ($\Vert B\Vert ={\sup }_{x\ne 0}\Vert Bx\Vert /\Vert x\Vert$)

A vector norm on a finite-dimensional real or complex vector space induces the displayed matrix norm on its linear operators. It is the corresponding operator norm, and the supremum can equivalently be taken over the unit sphere. Applying the length bound twice proves submultiplicativity: $\Vert BCx\Vert \le \Vert B\Vert \Vert C\Vert \Vert x\Vert$, hence $\Vert BC\Vert \le \Vert B\Vert \Vert C\Vert$.

For each row, the weighted Cauchy-Schwarz inequality gives $|(Bx{)}_i{|}^2\le ({\sum }_j|b_{ij}|)({\sum }_j|b_{ij}||x_j{|}^2)$. Summing over rows bounds the result by the maximum absolute row sum times the maximum absolute column sum times $\Vert x{\Vert }_2^2$. Taking square roots and the supremum proves the displayed induced matrix norm inequality. It gives a Euclidean estimate directly from entries without computing singular values.

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.