# Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower bound on the smallest eigenvalue of a finite sum of random positive semidefinite (PSD) matrices.

For PSD matrices, the eigenvalues are the same as the singular values. My question is:

Is there any tool, or any other special cases (apart from the PSD case) for which there is a tool, to obtain lower bounds on the singular values of a finite sum of random matrices, or random symmetrical matrices?

• Crossposted on Math.SE. Please do not crosspost the same question, on mutliple StackExchange sites at once. Otherwise, welcome! – Niel de Beaudrap Jan 5 '15 at 13:34
• This question is interesting, but potentially borderline for being on-topic: Chernoff bounds and bounds on singular values are clearly of interest in computer science (as are algorithms to compute them), but analytic upper/lower bounds on strictly mathematical objects would seem more a proper subject of mathematics. Perhaps this would be better answered on Math.SE. – Niel de Beaudrap Jan 5 '15 at 13:36
• or also MathOverflow – vzn Jan 8 '15 at 16:11