Let's say I have a block matrix of the form
$$X = \begin{bmatrix} A & B\\ B^T & C\end{bmatrix}$$
where $A$, $C$, and $X$ are all positive definite. I have bounds on both the minimum and maximum eigenvalues of $A$, $C$, and $X$. I am wondering what I can say about the maximum singular value of $B$.
Best Answer
Just a quick lazy answer.
By the interlacing property of Schur complements, for a vector $v$ with unit norm one has $\lambda_{\min}(X) \leq \lambda_{\min}(A-B C^{-1}B^T) \leq v^TAv - v^TB C^{-1} B^T v \leq \lambda_{\max}(A) - \frac1{\lambda_{\max}(C)}\|B^T v\|^2$, which gives the bound $$ (\sigma_{\max}(B))^2 \leq (\lambda_{\max}(A) - \lambda_{\min}(X))\lambda_{\max}(C). $$ I hope I'm not mixing up any min/max here, but in any case you get the idea of the reasoning.
It should be possible to find examples in which all equalities hold. Probably a similar inequality can be set up for the other direction, too ($\sigma_{\max}(B) \geq \sigma_{\min}(B) \geq \dots$).