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.

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References

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Bulletin of the Iranian Mathematical Society
Volume 43, Issue 6
November 2017
Pages 2057-2066

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History

  • Receive Date: 23 May 2016
  • Revise Date: 23 December 2016
  • Accept Date: 24 December 2016

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