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Maximum Principles for Normal Matrices
Hydrological Service, Jerusalem, Israel
- 1 Hydrological Service, Jerusalem, Israel
Advances in Linear Algebra & Matrix Theory·Volume 09 (2019)·Pages 73–81·Published 9 July 2019·DOI10.4236/alamt.2019.93005
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Abstract
Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.
KeywordsKy Fan Maximum PrincipleNormal MatricesExtended Maximum PrincipleUnitarily Invariant Norms
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