For $A \in \mathbb R^{m \times n}, m \geq n, x \in \mathbb R^{n \times 1}$, how to prove $\min\limits_{x \neq 0} \frac{\|Ax\|_2}{\|x\|_2}=\sigma_n$? As I'm new to SVD, can anyone help me in solving this problem?
Proving that $\min\limits_{x \neq 0} \frac{\|Ax\|_2}{\|x\|_2}=\sigma_n$
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linear-algebra
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0Do you know the corresponding proof for $\max$ and $\sigma_1$? – 2011-09-17
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0Im sorry. i dont know that proof. Proving any one of them is better – 2011-09-17
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0What have you tried and where are you stuck? If you substitute $A = U \Sigma V^T$ where the right-hand side is the SVD of $A$, then what is an equivalent form of the problem? – 2011-09-17