Singular value decomposition inequality

Question

Assume $A$ is nonsingular with SVD i.e \begin{equation} A = U \Sigma V^{T} \end{equation}

$U,V$ orthogonal.

$\Sigma$ is a diagonal matrix with non-negative entries $ \Sigma= diag (\sigma_{1}, \sigma_{2}, . . . , \sigma_{r})$ and $\sigma_1 ≥ \sigma_2 ≥\ldots ≥ \sigma_r> 0$ are the positive singular values of $A$.

Prove that

$\sigma_{n} \| x \| _{2} \leq \|Ax\|_{2} ≤ \sigma_{1}\|x\|_{2}$

Thanks!

Answer 1

You need to notice that, since $U $ and $V $ are orthogonal, $$\|Ax\|_2=\|\Sigma (V^Tx)\|_2\ \ \ \text {and } \ \ \|V^Tx\|_2=\|x\|_2 $$