Bound quotient of maximal and minimal singular values

Question

I am working on a problem where the following quantity emerges: $$ \frac{\sigma_{\text{max}}(J-M)}{\sigma_{\text{min}}(J+M)} $$ where $J$ is the canonical symplectic matrix, $M^T=M$ and $M$ is positive semi-definite.

I want to bound from above this quantity, which seems to be specific enough to try to get a good bound.

For the moment I have tried the simple approach that follows $$ \frac{\sigma_{\text{max}}(J-M)}{\sigma_{\text{min}}(J+M)}\leq \frac{1+\|M\|_2}{1-\|M\|_2} $$ where the last bound comes from an estimate in lower bound on the minimum singular value of $\underline{\sigma} (A+B)$ . Do you see a better solution?

Answer 1

There is a result from Bhatia's Matrix Analysis that might be helpful here:

Theorem VI.2.3: Let $A$ be Hermitian, and $B$ skew-Hermitian. Let their eigenvalues be arranged so that $$ |\alpha_1| \geq \cdots \geq |\alpha_n| \quad \text{and} \quad |\beta_1| \geq \cdots \geq |\beta_n| $$ Let $T = A + B$, and let $s_j$ be the singular values of $T$. Then the following majorization relations are satisfied: $$ \{|\alpha_j + \beta_{n - j+1}|^2\}_j \prec \{s_j^2\}_j\\ \left\{\frac 12(s_j^2 + s_{n - j + 1}^2) \right\}_j \prec \{|\alpha_j + \beta_{j}|^2\}_j $$

Majorization is explained on this wiki page. For our case, we note that $J$ is skew-symmetric with eigenvalues $\pm i$. With the second majorization condition, we have $$ \frac 12 \left(\sigma_{\max}^2([-M] + J) + \sigma_{\min}^2([-M] + J)\right) \leq 1 + \|M\|_2^2 \implies $$ \begin{align} \sigma_{\max}^2(J-M) &\leq 2(1 + \sigma_{\max}^2(M)) - (1 + \sigma_{\min}^2(M)) \\ & = 1 + \left[2 \sigma_{\max}^2(M)\ - \sigma_\min^2(J - M)\right], \end{align} which might yield a tighter upper bound on the numerator.