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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?

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    What 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

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