Diagonals and Partial Diagonals of Sum of Matrices

Li, Chi-Kwong ; Poon, Yiu-Tung

Canadian journal of mathematics, Tome 54 (2002) no. 3, pp. 571-594

Résumé

Given a matrix $A$ , let $\mathcal{O}\left( A \right)$ denote the orbit of $A$ under a certain group action such as (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) $O\left( m \right)\otimes O\left( n \right)$ 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) $U(n)$ acting on $n\,\times \,n$ complex symmetric or skew-symmetric matrices $A$ by $U\,*\,A\,=\,UA{{U}^{t}}$ , (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.

Détail

DOI : 10.4153/CJM-2002-020-1

Mots-clés : 15A42, 15A18, orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular values

Bibliographie

[1] [1] Fulton, W., Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) (3) 37 (2000), 209–249 Google Scholar

[2] [2] Horn, A., Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math. 76 (1954), 620–630. Google Scholar

[3] [3] Horn, A., Eigenvalues of sums of Hermitian matrices. Pacific J. Math. 12 (1962), 225–241. Google Scholar

[4] [4] Horn, R. A. and Johnson, C. R., Topics in Matrix Analysis. Cambridge University Press, 1991. Google Scholar

[5] [5] Li, C. K., Matrices with some extremal properties. Linear Algebra Appl. 101 (1988), 255–267. Google Scholar

[6] [6] Marshall, A. W. and Olkin, I., Inequalities: The Theory of Majorizations and Its Applications. Academic Press, 1979. Google Scholar

[7] [7] Miranda, H. F., Singular values, diagonal elements, and extreme matrices. Linear Algebra Appl. 305 (2000), 151–159. Google Scholar

[8] [8] Miranda, H. F. and Thompson, R. C., A supplement to the von Neumann trace inequality for singular values. Linear Algebra Appl. 248 (1996), 61–66. Google Scholar

[9] [9] Mirsky, L., Matrices with prescribed characteristic roots and diagonal elements. J. London Math. Soc. 33 (1958), 14–21. Google Scholar

[10] [10] O’Shea, L. and Sjamaar, R., Moment maps and Riemannian symmetric pairs. Math. Ann. 317 (2000), 415–457. Google Scholar

[11] [11] Schur, I., Über eine Klasse von Mittelbildungen mit Anwendungen die Determinanten-Theorie Sitzungsber. Berlin. Math. Gesellschaft 22 (1923), 9–20. (See also, Issai Schur Collected Works (eds. A. Brauer and H. Rohrbach), Vol. II, Springer-Verlag, Berlin, 1973, 416–427.) Google Scholar

[12] [12] Sing, F. Y., Some results on matrices with prescribed diagonal elements and singular values. Canad. Math. Bull. 19 (1976), 89–92. Google Scholar

[13] [13] Tam, T. Y., Kostant's convexity theorem and the compact classical groups. Linear and Multilinear Algebra 43 (1997), 87–113. Google Scholar

[14] [14] Tam, T. Y., Partial superdiagonal elements and singular values of a complex skew-symmetric matrix. SIAM J. Matrix Anal. Appl. 19 (1998), 737–754. Google Scholar

[15] [15] Tam, T. Y., On Lei, Miranda, and Thompson's result on singular values and diagonal elements. Linear Algebra Appl. 272 (1998), 91–101. Google Scholar

[16] [16] Tam, T. Y., A Lie theoretic approach to Thompson's theorems on singular values-diagonal elements and some related results. J. London Math. Soc. 60 (1999), 431–448. Google Scholar

[17] [17] Thompson, R. C., Principal submatrices IX: Interlacing inequalities for singular values of submatrices. Linear Algebra Appl. 5 (1972), 1–12. Google Scholar

[18] [18] Thompson, R. C., Singular values, diagonal elements, and convexity. SIAM J. Appl. Math. 32 (1977), 39–63. Google Scholar

[19] [19] Thompson, R. C., Singular values and diagonal elements of complex symmetric matrices. Linear Algebra Appl. 26 (1979), 65–106. Google Scholar

[20] [20] Thompson, R. C., The diagonal torus of a matrix under special unitary equivalence. SIAM J. Matrix Anal. Appl. 15 (1994), 968–973. Google Scholar