Generalized Finite Algorithms for Constructing Hermitian Matrices with Prescribed Diagonal and Spectrum

In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N...

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Vydáno v:SIAM journal on matrix analysis and applications Ročník 27; číslo 1; s. 61 - 71
Hlavní autoři: Dhillon, Inderjit S., Heath, Robert W., Sustik, Mátyás A., Tropp, Joel A.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2005
Témata:
Algebra
Algorithms
Eigenvalues
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
inverse eigenvalue problem
Spectral properties
Singular value
Hermitian matrix
Plane rotation
plane rotations
Hermitian matrices
Algorithm
Inverse problem
Spectrum
Schur Horn theorem
15A57
Diagonal matrix
Schur-Horn theorem
Majoration
finite-step algorithms 15A18
Eigenvalue problem
majorization
65F18
ISSN:0895-4798, 1095-7162
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Popis
Shrnutí:In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N is the dimension of the matrix. Both the Bendel--Mickey and the Chan--Li algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as $\mtx{X^\adj X}$, we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0895-4798
1095-7162
DOI:10.1137/S0895479803438183