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Diagonals and Partial Diagonals of Sum of Matrices

Published online by Cambridge University Press:  20 November 2018

Chi-Kwong Li  and
Yiu-Tung Poon
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187, USA, email: [email protected]
Yiu-Tung Poon
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA, email: [email protected]
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Abstract

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Given a matrix $A$, let $\mathcal{O}\left( A \right)$ denote the orbit of $A$ under a certain group action such as

  1. (1) $U\left( m \right)\,\otimes U\left( n \right)$ acting on $m\,\times \,n$ complex matrices $A$ by $(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,

  2. (2) $O\left( m \right)\otimes O\left( n \right)$ or $\text{SO(}m\text{)}\,\otimes \,\text{SO(}n\text{)}$ acting on $m\,\times \,n$ real matrices $A$ by $(U,\,V)\,*\,A\,=\,UA{{V}^{t}}$,

  3. (3) $U(n)$ acting on $n\,\times \,n$ complex symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$,

  4. (4) $O(n)$ or $\text{SO(n)}$ acting on $n\,\times \,n$ real symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$.

Denote by

1

$$\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})\,=\,\{{{X}_{1\,}}+\cdot \cdot \cdot +\,{{X}_{k}}\,:\,{{X}_{i}}\,\in \,\mathcal{O}({{A}_{i}}),i\,=\,1,\ldots ,k\}$$

the joint orbit of the matrices ${{A}_{1}},\ldots ,{{A}_{k}}$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$, i.e., the set of vectors $({{d}_{1}},\ldots {{d}_{r}})$ whose entries lie in the $(1,\,{{j}_{1}}),\ldots ,(r,\,{{j}_{r}})$ positions of a matrix in $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ for some distinct column indices ${{j}_{1}},\ldots ,{{j}_{r}}$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of ${{A}_{1}},\ldots ,{{A}_{k}}$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}({{A}_{1}},\ldots ,{{A}_{k}})$ of complex Hermitian matrices under the action of unitary similarities are also discussed.

Keywords

15A4215A18orbitgroup actionsunitaryorthogonalHermitian(skew-)symmetric matricesdiagonalsingular values
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 3 , 01 June 2002 , pp. 571 - 594
Copyright
Copyright © Canadian Mathematical Society 2002

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