Submodular spectral functions of principal submatrices of a hermitian matrix, extensions and applications

Abstract

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 $I\mapsto \mathrm{tr}f(A[I])$ is supermodular, meaning that $\mathrm{tr}f(A[I])+\mathrm{tr}f(A[J])\le \mathrm{tr}f(A[I\cup J])+\mathrm{tr}f(A[I\cap J])$, where $A[I]$ denotes the $I\times I$ principal submatrix of $A$. We discuss extensions to self-adjoint operators on infinite dimensional Hilbert space and to $M$-matrices. We discuss an application to CUR approximation of nonnegative hermitian matrices.

16 pages