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

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 $I\mapsto {\rm tr} f(A[I])$ 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 to self-adjoint operators on infinite dimensional Hilbert space and to $M$-matrices. We discuss an application to CUR approximation of nonnegative hermitian matrices.

💡 Research Summary

The paper builds on the well‑known multiplicative submodularity of principal determinants of a non‑negative‑definite Hermitian matrix and extends the phenomenon to a broad class of spectral functions. The authors consider a Hermitian matrix (A\in\mathbb{C}^{n\times n}) and, for any index set (I\subseteq{1,\dots,n}), denote by (A