%0 Journal Article
%T Cut-norms and spectra of matrices
%A Vladimir Nikiforov
%J Mathematics
%D 2009
%I arXiv
%X One of the aims of this paper is to solve an open problem of Lovasz about relations between graph spectra and cut-distance. The paper starts with several inequalities between two versions of the cut-norm and the two largest singular values of arbitrary complex matrices, exteding, in particular, the well-known graph-theoretical Expander Mixing Lemma and giving a hitherto unknown converse of it. Next, cut-distance is defined for Hermitian matrices, and, separately, for arbitrary complex matrices; using these extensions, we give upper bounds on the difference of corresponding eigenvalues and singular values of two matrices, thus solving the problem of Lovasz. Finally, we deduce a spectral sampling theorem, which informally states that almost all principal submatrices of a real symmetric matrix are spectrally similar to it.
%U http://arxiv.org/abs/0912.0336v1