Proof of matrix norm property: submultiplicativity

proof-writing matrices normed-spaces

If you have a norm on $K^n$ and if you define a matrix norm (the induced matrix norm)

$$\lVert A\rVert=\sup\limits_{\lVert x\rVert =1}\{\lVert Ax\rVert :x\in K^n\}$$

then one can readily check $$\lVert Ax\rVert \leqslant \lVert A\rVert \lVert x\rVert $$

There is no problem that $A,B$ aren't square, as long as $AB$ makes sense. In such a case

$$\lVert ABx\rVert \leqslant \lVert A\rVert \lVert Bx\rVert \leqslant \lVert A\rVert \lVert B\rVert \lVert x\rVert $$

You can check Rudin's Princples Chapter 9, which has this on the very first few pages.

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