24 Norm of a matrix

(ii) By (i) we have

$${\parallel z+w\parallel }^2=⟨z+w,z+w⟩$$

$$=⟨z,z⟩+⟨z,w⟩+⟨w,z⟩+⟨w,w⟩$$

$$={\parallel z\parallel }^2+2\mathrm{\Re }⟨z,w⟩+{\parallel w\parallel }^2$$

$$\le {\parallel z\parallel }^2+2\parallel z\parallel \parallel w\parallel +{\parallel w\parallel }^2.$$

Suppose that $A$ and $B$ are $(n\times n)$ complex matrices, and that $V={𝐂}^{n\times 1}$. Then $A$ operates on $V={𝐂}^{(n\times 1)}$ column vector and $A:V\to V:$ $v\mapsto Av$ by multiplication on the left. The norm of a $(n\times n)$ matrix is

$$\parallel A\parallel =sup\{\parallel Av\parallel :v\in V;\parallel v\parallel \le 1\}.$$