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Let $\|\cdot\|_2·$ be the norm of $C^n$ defined by $\|x\|_2=\left(|x_1|^2+|x_2|^2+...+|x_n|^2\right)^{1/2}$. Prove that for the subordinate matrix norm $∥⋅∥_2$ must be $\|A\|_2=\max\{ \sqrt{|λ|}:λ \text{ is an eigenvalue of } AA^∗ \}$ where $A^*$ is the conjugate transpose of $A$.

Eugene Zhang
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hermanos sierra
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1 Answers1

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Hint:

Since $AA^*$ is Hermitian, it is unitarily similar to a diagonal matrix, i.e. $$ AA^*=U\operatorname{diag}(|\lambda_1|,\cdots,|\lambda_n|)U^* $$ where $\lambda_i$ are eigenvalues of $AA^*$ and $UU^*=I$. Then $$ \|A\|_2^2=\sup_{\|x\|=1}\|Ax\|^2=\sup_{\|x\|=1}\sum_{1\leqslant i\leqslant n}|U^*x|_{i}^2|\lambda_i|=\max_{1\leqslant i\leqslant n}{(|\lambda_i|)} $$

Eugene Zhang
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