0.0.1. Def:consistent_norm

consistent norm(両立するノルム)

Let $K$ be a field of either real or complex numbers, $\parallel \cdot \parallel$ be a matrix norm on $K^{m\times n}$ is called consistent with a vector norm $\parallel \cdot {\parallel }_a$ on $K^n$ and vector norm $\parallel \cdot {\parallel }_b$ on $K^m$ if:

$\parallel Ax{\parallel }_b\le \parallel A\parallel \parallel x{\parallel }_a\forall A\in K^{m\times n},x\in K^n$

0.0.2. Th:Frobenius_norm_is_consistent

フロベニウスノルムは両立するノルム

Let $K$ be a field of either real or complex numbers, $\parallel \cdot \parallel$ be Frobenius norm on $K^m$,$K^n$, or $K^{m\times n}$. Then $\parallel \cdot \parallel$ is consistent norm.

(proof)

$A\in K^{m\times n}$と$x\in K^n$を任意にとる。この時、コーシー・シュワルツの不等式を用いると $$\begin{array}{cc}\parallel Ax{\parallel }^2& =\sum _{j=1}^m\left({\left|\sum _{i=1}^m{a}_{ji}{x}_i\right|}^2\right)\\ & =\sum _{j=1}^m\sum _{i=1}^m|a_{ji}{x}_i{|}^2\\ & =\sum _{j=1}^m{\left(\sum _{i=1}^m|a_{ji}{x}_i|\right)}^2\\ & =\sum _{j=1}^m(\left(\sum _{i=1}^n|a_{ji}{|}^2\right)\cdot \left(\sum _{i=1}^n|x_i{|}^2\right))\\ & =\parallel A{\parallel }^2\parallel x{\parallel }^2\end{array}$$