Singular value decomposition of matrix products

Question

For $A ∈ C^{m×n}$ and $B ∈ C^{n×k}$ with $m ≥ k$ and $n ≥ k$, how can I prove that

$\sigma_k(AB) ≤ \sigma_1(A) \sigma_k(B)$

?

Answer 1

Sounds like nonsense to me. Try $m=n=k=2$, $$A = B = \left[\begin{array}{cc}2 & 0\\0 & 0\end{array}\right].$$

Maybe you meant $\sigma_k(AB) \geq \sigma_1(A)\sigma_k(B)?$ Here's a hint: what do you know about bounds $c$ and $C$ on the operator norm $$\|Ax\| \leq C\|x\|$$ $$\|Ax\| \geq c\|x\|$$ in terms of the singular values of $A$?