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Off-diagonal submatrices of a Hermitian matrix

Authors: Chi-Kwong Li and Yiu-Tung Poon
Journal: Proc. Amer. Math. Soc. 132 (2004), 2849-2856
MSC (2000): Primary 15A18, 15A42
Published electronically: June 2, 2004
MathSciNet review: 2063102
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Abstract: A necessary and sufficient condition is given to a $p\times q$ complex matrix $X$ to be an off-diagonal block of an $n\times n$ Hermitian matrix $C$ with prescribed eigenvalues (in terms of the eigenvalues of $C$ and singular values of $X$). The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton. Some interesting geometrical properties of the set $S$ of all such matrices are derived from the main result. These results improve earlier ones that only give partial information for the set $S$.

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Additional Information

Chi-Kwong Li
Affiliation: Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795

Yiu-Tung Poon
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011

Keywords: Hermitian matrices, singular values, eigenvalues, Littlewood-Richardson rules
Received by editor(s): February 18, 2002
Received by editor(s) in revised form: October 24, 2002
Published electronically: June 2, 2004
Additional Notes: The first author’s research was partially supported by an NSF grant
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society