Submodular spectral functions of principal submatrices of a hermitian matrix, extensions and applications
| Autor: | Friedland, S., Gaubert, S. |
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| Rok vydání: | 2010 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 16 pages |
| Databáze: | arXiv |
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