Diagonals and Partial Diagonals of Sum of Matrices

摘要

Given a matrix $A$, let $mathcal{O}(A)$ denote the orbit of $A$ under acertain group action such as egin{enumerate}[(4)]item[(1)] $U(m) otimes U(n)$ acting on $m imes n$ complex matrices$A$ by $(U,V)*A = UAV^t$, item[(2)] $O(m) otimes O(n)$ or $SO(m) otimes SO(n)$ acting on $m imes n$ real matrices $A$ by $(U,V)*A = UAV^t$,item[(3)] $U(n)$ acting on $n imes n$ complex symmetric orskew-symmetric matrices $A$ by $U*A = UAU^t$, item[(4)] $O(n)$ or $SO(n)$ acting on $n imes n$ real symmetric orskew-symmetric matrices $A$ by $U*A = UAU^t$.end{enumerate}Denote by$$mathcal{O}(A_1,dots,A_k) = {X_1 + cdots + X_k : X_i inmathcal{O}(A_i), i = 1,dots,k} $$the joint orbit of the matrices $A_1,dots,A_k$. We study the set ofdiagonals or partial diagonals of matrices in $mathcal{O}(A_1,dots,A_k)$,{it i.e.}, the set of vectors $(d_1,dots,d_r)$ whose entries liein the $(1,j_1),dots,(r,j_r)$ positions of a matrix in $mathcal{O}(A_1,dots,A_k)$ for some distinct column indices $j_1,dots,j_r$. In manycases, complete description of these sets is given in terms of theinequalities involving the singular values of $A_1,dots,A_k$. Wealso characterize those extreme matrices for which the equality caseshold. Furthermore, some convexity properties of the joint orbits areconsidered. These extend many classical results on matrixinequalities, and answer some questions by Miranda. Related resultson the joint orbit $mathcal{O}(A_1,dots,A_k)$ of complexHermitian matrices under the action of unitary similarities arealso discussed.