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Suppose $A\in\mathbb{R}^{m\times n}$ and consider the matrix $2$-norm $$\|A\|_{2} = \max_{\|x\|_{2} = 1}\|Ax\|_{2}$$ Show that $\|A\|_{2} \geq \|A_1\|_{2}$ where $$A = \begin{pmatrix} A_1\\ A_2\\ \end{pmatrix}$$ $m = m_1 + m_2$, $A_1\in\mathbb{R}^{m_1\times n}$ and $A_2\in\mathbb{R}^{m_2\times n}$

Attempted solution - There exists $x_1\in\mathbb{R}^n$ such that $$\|A_1 x_1\|_{2} = \|A_1\|_{2}$$ Therefore, $$\|A\|_{2}^{2} \geq \|Ax_1\|_{2}^{2} = \|A_1 x_1\|_{2}^{2} + \|A_2 x_1\|_{2}^{2} \geq \|A_1 x_1\|_{2}^{2} = \|A_1\|_{2}^{2}$$

I feel like this proof is not that elegant enough. If anyone has any suggestions on this that would be great.

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  • $\begingroup$ The first inequality should read $\|A\|_2^2\ge \|Ax_1\|_2^2$. What you could do to "improve" your answer is to avoid using compactness... but I don't see your proof as "inelegant" $\endgroup$
    – b00n heT
    May 13, 2016 at 13:13
  • $\begingroup$ Got it thanks for that $\endgroup$
    – Wolfy
    May 13, 2016 at 13:28

1 Answer 1

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Here's the proof I like: $$ \|A\|^2 = \max_{\|x\| = 1} \|Ax\|^2 = \max_{\|x_1\|^2 + \|x_2\|^2 = 1} \|Ax_1\|^2 + \|Ax_2\|^2 \geq \\ \max_{\|x_1\| = 1, x_2 = 0} \|A_1x_1\|^2 + \|A_2x_2\|^2 = \max_{\|x_1\| = 1} \|A_1x_1\|^2= \|A_1\|^2 $$ The second max is necessarily smaller since $$ \{(x_1,x_2) : \|x_1\|^2 + \|x_2\|^2 =1 \} \supseteq \{(x_1,x_2): \|x_1\| = 1, x_2 = 0\} $$

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  • $\begingroup$ Or, you could always apply your previous result $\endgroup$ May 13, 2016 at 13:25
  • $\begingroup$ Nice, by the way unrelated but is it better to have a stronger answer to question ratio on MSE? $\endgroup$
    – Wolfy
    May 13, 2016 at 13:30
  • $\begingroup$ It's not. The only thing that "matters" is how many points you get. That being said, when you have a lot of answers, people tend to assume you know what you're talking about ;) $\endgroup$ May 13, 2016 at 13:32
  • $\begingroup$ Nice, I will have to answer more questions after I get passed my quals. Such a grind... $\endgroup$
    – Wolfy
    May 13, 2016 at 13:33
  • $\begingroup$ Do it if you enjoy it. If anything, I'd say that hunting for linear algebra questions (click the tag to find the related questions) was good prep for my quals, but doing old test questions is probably better. At first, this was an excuse for me to get better at TeXing. $\endgroup$ May 13, 2016 at 13:35

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