JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 27, Number 2, April 2014, Pages 293–338 S 0894-0347(2013)00771-7 Article electronically published on May 8, 2013
INVERTIBILITY OF RANDOM MATRICES: UNITARY AND ORTHOGONAL PERTURBATIONS
MARK RUDELSON AND ROMAN VERSHYNIN In memory of Joram Lindenstrauss
Contents 1. Introduction 294 1.1. The smallest singular values of random matrices 294 1.2. The main results 294 1.3. A word about the proofs 296 1.4. An application to the Single Ring Theorem 296 1.5. Organization of the paper 298 1.6. Notation 298 2. Strategy of the proofs 298 2.1. Unitary perturbations 298 2.2. Orthogonal perturbations 299 3. Unitary perturbations: Proof of Theorem 1.1 302 3.1. Decomposition of the problem; local and global perturbations 302 3.2. Invertibility via quadratic forms 304 3.3. When the denominator is small 306 3.4. When the denominator is large and M is small 306 3.5. When M is large 308 3.6. Combining the three cases 309 4. Orthogonal perturbations: Proof of Theorem 1.3 310 4.1. Initial reductions of the problem 310 4.2. Local perturbations and decomposition of the problem 310 4.3. When a minor is well invertible: Going 3 dimensions up 313 4.4. When all minors are poorly invertible: Going 1 + 2 dimensions up 320 4.5. Combining the results for well-invertible and poorly invertible minors 322 5. Application to the Single Ring Theorem: Proof of Corollary 1.4 323 Appendix A. Orthogonal perturbations in low dimensions 327 A.1. Remez-type inequalities 327 A.2. Vanishing determinant 328 A.3. Proof of Theorem 4.1 330 Appendix B. Some tools used in the proof of Theorem 1.3 334 B.1. Small ball probabilities 334 B.2. Invertibility of random Gaussian perturbations 334 B.3. Breaking complex orthogonality 336
Received by the editors June 22, 2012 and, in revised form, January 30, 2013. 2010 Mathematics Subject Classification. Primary 60B20. The first author was partially supported by NSF grant DMS 1161372. The second author was partially supported by NSF grant DMS 1001829.