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ELA

SHORT PROOFS OF THEOREMS OF MIRSKY AND HORN ON

DIAGONALS AND EIGENVALUES OF MATRICES

∗

ERIC A. CARLEN† AND ELLIOTT H. LIEB‡

Abstract. A theorem of Mirsky provides necessary and sufficient conditions for the existence of anN-square complex matrix with prescribed diagonal entries and prescribed eigenvalues. A simple inductive proof of this theorem is given.

Key words. Majorization, Eigenvalues, Prescribed diagonals.

AMS subject classifications.15A42, 15A51.

∗Received by the editors May 13, 2009. Accepted for publication July 24, 2009. Handling Editor:

Roger A. Horn.

†Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghuysen Road,

Piscat-away, NJ 08854-8019. Work partially supported by U.S. National Science Foundation grant DMS 06-00037.

‡ Departments of Mathematics and Physics, Jadwin Hall, Princeton University, P. O. Box 708,

Princeton, NJ 08542. Work partially supported by U.S. National Science Foundation grant PHY 06-52854.

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 438-441, July 2009

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