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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.22 n.2 Antofagasta ago. 2003

http://dx.doi.org/10.4067/S0716-09172003000200003 

Proyecciones
Vol. 22, N o 2, pp. 127-134, August 2003.
Universidad Católica del Norte
Antofagasta - Chile

DIAGONALS AND EIGENVALUES OF SUMS
OF HERMITIAN MATRICES. EXTREME CASES*

HÉCTOR MIRANDA
Universidad de Bío - Bío, Concepción – Chile

Abstract

There are well known inequalities for Hermitian matrices A and
B that relate the diagonal entries of A+B to the eigenvalues of A and
B. These inequalities are easily extended to more general inequalities
in the case where the matrices A and B are perturbed through con-gruences
of the form UAU*+ V BV *; where U and V are arbitrary
unitary matrices, or to sums of more than two matrices. The extremal
cases where these inequalities and some generalizations become equal-ities
are examined here.

Key words. Hermitian matrix, eigenvalues, diagonal elements.

AMS Subject Classification. 15A18, 15A42.

References

[1] K. Fan, On a theorem of Weyl concerning eigenvalues of linear trans-formations I, Proc. Nat. Acad. Sci. U.S.A. 35: pp. 52-655, (1949).        [ Links ]

[2] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge U. P., New York, (1985).        [ Links ]

[3] C. K. Li, Matrices with some extremal properties, Linear Algebra Appl.101: pp. 255-267, (1988).        [ Links ]

[4] C. K. Li and Y. T. Poon, Diagonal and partial diagonals of sums of matrices, Canad. J. Math. 54: pp. 571-594, (2002).        [ Links ]

[5] H. Miranda, Optimality of the trace of a product of matrices, Proyecciones Revista de Matemática 18, 1: pp. 71-76, (1999).        [ Links ]

[6] H. Miranda, Singular values, diagonal elements, and extreme matrices, Linear Algebra Appl. 305: pp. 151-159, (2000).        [ Links ]

[7] H. Miranda and R. C. Thompson, A supplement to the von Neumann trace inequality for singular values, Linear Algebra Appl. 248: pp. 61-66, (1996).        [ Links ]

[8] I. Schur, Uber eine klasse von mittelbildungen mit anwendungen auf die determinantentheorie, Sitzungsber. Berliner Math. Ges. 22: pp. 9-20, (1923).        [ Links ]

[9] R. C. Thompson, Singular values, diagonal elements, and convexity,SIAM J. Appl. Math. 32: pp. 39-63, (1977).        [ Links ]Received : January 2003.
Universidad del Bío-Bío

Héctor Miranda
Departamento de Matemática
Casilla 5-C
Concepción
Chile
e-mail : hmiranda@ubiobio.cl


*Partially funded by Fondecyt 1010487

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