# To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and that the inequality may be strict.

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What is $|| A ||$ here ? (there are several matrix norms) –  Ewan Delanoy Jan 25 '13 at 10:23
unit norm matrix –  aneps Jan 25 '13 at 10:49
You mean ${\sf sup}\frac{||Av||}{||v||}$ ? –  Ewan Delanoy Jan 25 '13 at 10:56
whats definition of this norm ? –  Maisam Hedyelloo Jan 25 '13 at 10:58
@EwanDelanoy: Yes –  aneps Jan 25 '13 at 11:02
If you mean the operator norm, here is a hint: $\|A\|=\max_{\|x\|=1}\|Ax\|=\max_{\|y\|=1}\|y^\ast A\|$. Now take $x$ and $y$ as vectors from the canonical bases of $\mathbb{C}^q$ and $\mathbb{C}^p$.