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Article Dans Une Revue Linear Algebra and its Applications Année : 2013

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

Résumé

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.

Dates et versions

hal-00940794 , version 1 (02-02-2014)

Identifiants

Citer

S. Friedland, S. Gaubert. Submodular spectral functions of principal submatrices of a hermitian matrix, extensions and applications. Linear Algebra and its Applications, 2013, 438 (10), pp.3872-3884. ⟨10.1016/j.laa.2011.11.021⟩. ⟨hal-00940794⟩
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