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

摘要

We extend the multiplicative submodularity of the principal determinants of anonnegative definite hermitian matrix to other spectral functions. We show thatif $f$ is the primitive of a function that is operator monotone on an intervalcontaining the spectrum of a hermitian matrix $A$, then the function $I\mapsto{\rm tr} f(A[I])$ is supermodular, meaning that ${\rm tr} f(A[I])+{\rm tr}f(A[J])\leq {\rm tr} f(A[I\cup J])+{\rm tr} f(A[I\cap J])$, where $A[I]$denotes the $I\times I$ principal submatrix of $A$. We discuss extensions toself-adjoint operators on infinite dimensional Hilbert space and to$M$-matrices. We discuss an application to CUR approximation of nonnegativehermitian matrices.