Singular values of convex functions of matrices

Document Type: Research Paper

Author

Vadi Park Sit.‎, ‎Gulvatan Sok.‎, ‎Yazir‎, ‎Selçuklu‎, ‎42250‎, ‎Konya‎, ‎Turkey.

Abstract

‎Let $A_{i},B_{i},X_{i},i=1,\dots,m,$ be $n$-by-$n$ matrices such that $‎\sum_{i=1}^{m}\left\vert A_{i}\right\vert ^{2}$ and $‎\sum_{i=1}^{m}\left\vert B_{i}\right\vert ^{2}$  are nonzero matrices and each $X_{i}$ is‎ ‎positive semidefinite‎. ‎It is shown that if $f$ is a nonnegative increasing ‎convex function on $\left[ 0,\infty \right) $ satisfying $f\left( 0\right)‎ ‎=0 $‎, ‎then 
$$‎2s_{j}\left( f\left( \frac{\left\vert \sum_{i=1}^{m}A_{i}^{\ast‎ ‎ }X_{i}B_{i}\right\vert }{\sqrt{\left\Vert \sum_{i=1}^{m}\left\vert‎
‎ A_{i}\right\vert ^{2}\right\Vert \left\Vert \sum_{i=1}^{m}\left\vert‎
‎ B_{i}\right\vert ^{2}\right\Vert }}\right) \right) \leq s_{j}\left( \oplus‎
‎_{i=1}^{m}f\left( 2X_{i}\right) \right)‎$$
‎for $j=1,\ldots,n$‎. ‎Applications of our results are given.

Keywords

Main Subjects


H. Albadawi, Singular value and arithmetic-geometric mean inequalities for operators, Ann. Funct. Anal. 3 (2012), no. 1, 10--18.

J.S. Aujla and F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217--233.

R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.

R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. 11 (1990), no. 2, 272--277.

R. Bhatia and F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Linear Algebra Appl. 308 (2000) 203--211.

K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations I, Proc. Natl. Acad. Sci. USA 35 (1994) 652--655.

O. Hirzallah, Singular values of convex functions of operators and the arithmetic-geometric mean inequality, J. Math. Anal. Appl. 433 (2016) 935--947.

R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985.

A.W. Marshall, I. Olkin and B.C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer Ser. Statist. Springer, 2nd edition, New York, 2011.

Y. Tao, More results on singular value inequalities of matrices, Linear Algebra Appl. 416 (2006) 724--729.

X. Zhan, Singular values of differences of positive semidefinite matrices, SIAM J. Matrix Anal. Appl. 22 (2000), no. 3, 819--823.

X. Zhan, Matrix Inequalities, Lecture Notes in Math. 1790, Springer-Verlag, Berlin, 2002.

L. Zou, An arithmetic-geometric mean inequality for singular values and its applications, Linear Algebra Appl. 528 (2017) 25--32.


Volume 43, Issue 6
November and December 2017
Pages 2057-2066
  • Receive Date: 23 May 2016
  • Revise Date: 23 December 2016
  • Accept Date: 24 December 2016