Diagonals and eigenvalues of sums of hermitian matrices. Extreme cases

Authors

  • Héctor Miranda Universidad de Bío – Bío.

DOI:

https://doi.org/10.4067/S0716-09172003000200003

Keywords:

Hermitian matrix, Eigenvalues, Diagonal elements.

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 congruences 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 equalities are examined here.

Author Biography

Héctor Miranda, Universidad de Bío – Bío.

Departamento de Matemática.

References

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

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

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

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

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

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

[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).

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

[9] R. C. Thompson, Singular values, diagonal elements, and convexity, SIAM J. Appl. Math. 32: pp. 39-63, (1977).

Published

2017-04-24

How to Cite

[1]
H. Miranda, “Diagonals and eigenvalues of sums of hermitian matrices. Extreme cases”, Proyecciones (Antofagasta, On line), vol. 22, no. 2, pp. 127-134, Apr. 2017.

Issue

Section

Artículos