2010/07/20 by Shmuel Friedland, Stéphane Gaubert, Friedland, S. +1 · 1 citation
Computer Science · Mathematics · #15A18 #15B57 #90C10 #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1007.3478
openalex publication_date 2010/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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↦ \rm tr f(A[I]) is supermodular, meaning that \rm tr f(A[I])+\rm tr f(A[J])≤ \rm tr f(A[I∪ J])+\rm tr f(A[I∩ J]), where A[I] denotes the I× 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.