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

Dhillon, Inderjit S. and Heath, Robert W., Jr. and Sustik, Mátyás A. and Tropp, Joel A. (2005) Generalized Finite Algorithms for Constructing Hermitian Matrices with Prescribed Diagonal and Spectrum. SIAM Journal on Matrix Analysis and Applications, 27 (1). pp. 61-71. ISSN 0895-4798. doi:10.1137/S0895479803438183.

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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.

Item Type:Article
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Tropp, Joel A.0000-0003-1024-1791
Additional Information:©2005 Society for Industrial and Applied Mathematics. Received by the editors November 21, 2003; accepted for publication (in revised form) by U. Helmke September 24, 2004; published electronically June 22, 2005. The research of this author [I.S.D.] was supported by NSF CAREER Award ACI-0093404.
Subject Keywords:Hermitian matrices; majorization; Schur-Horn theorem; inverse eigenvalue problem; plane rotations; finite-step algorithms
Issue or Number:1
Record Number:CaltechAUTHORS:DHIsiamjmaa05
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9036
Deposited By: Archive Administrator
Deposited On:22 Oct 2007
Last Modified:08 Nov 2021 20:55

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