You need to know: Square matrix with complex entries, eigenvalue of a square matrix, identity matrix I, conjugate transpose of matrix , unitary matrix (matrix such that ), group, unitary group (group of all unitary matrices with matrix multiplication as group operation), notation for probability, uniform distribution in the unitary group.
Background: For any matrix A, the eigenvalues of are real and non-negative. The square roots of these eigenvalues are called singular values of A. Let be the smallest singular value.
The Theorem: On 22nd June 2012, Mark Rudelson and Roman Vershynin submitted to arxiv and the Journal of the AMS a paper in which they proved the following result. Let D be an arbitrary fixed matrix, . Let U be a random matrix uniformly distributed in the unitary group . Then for all and some positive constants .
Short context: The smallest singular value of a square matrix A is inverse proportional to the norm of , and therefore provides a quantitative measure of invertibility of A. In earlier works (see here and here), Rudelson and Vershynin analyse invertibility of random matrices with independent entries. The Theorem did the same for random unitary perturbation of a fixed matrix D. It is important that the estimate in the Theorem does not depend on D.
Links: Free arxiv version of the original paper is here, journal version is here.
Theorems of the 21st century 