Exercise 1290Topology › Topology of Mn(ℝ)L1Added 4 Nov 2026

Submultiplicativity of matrix norms

#submultiplicative norms#operator norm#matrix norm

Statement

Define $\|A\|_{\mathrm{op}}=\sup_{\|x\|=1}\|Ax\|$ (for $\|\cdot\|$ the Euclidean norm on $\mathbb{R}^n$ ). Show that $\|AB\|_{\mathrm{op}}\le\|A\|_{\mathrm{op}}\|B\|_{\mathrm{op}}$ .

Hint

Write $\|ABx\|\le\|A\|_{\mathrm{op}}\|Bx\|\le\|A\|_{\mathrm{op}}\|B\|_{\mathrm{op}}\|x\|$ .

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Solution

Inequality. For every $x$ :
$\|ABx\|\le\|A\|_{\mathrm{op}}\|Bx\|\le\|A\|_{\mathrm{op}}\|B\|_{\mathrm{op}}\|x\|.$
Taking the supremum over $\|x\|=1$ : $\|AB\|_{\mathrm{op}}\le\|A\|_{\mathrm{op}}\|B\|_{\mathrm{op}}$ .

Conclusion. $\|\cdot\|_{\mathrm{op}}$ is a submultiplicative matrix norm.

Submultiplicativity of matrix norms

Statement

Solution

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