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Submodular spectral functions of principal submatrices of a hermitian matrix, extensions and applications
journal contributionposted on 2013-12-19, 00:00 authored by S. Friedland, S. Gaubert
We extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions. We show that if f is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix A, then the function I bar right arrow trf(A[I]) is supermodular, meaning that trf(A[I]) + trf(A[J]) <= trf(A[I boolean OR J]) + trf(A[I boolean AND J]), where A[I] denotes the I x I principal submatrix of A. We discuss extensions to self-adjoint operators on infinite dimensional Hilbert space and to M-matrices. We also discuss an application to CUR approximation of nonnegative hermitian matrices.
Publisher StatementNOTICE: This is the author’s version of a work that was accepted for publication in Linear Algebra and Its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Linear Algebra and Its Applications, Vol 438, Issue 10, 2013 DOI: 10.1016/j.laa.2011.11.021