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1. CJM 2009 (vol 62 pp. 109)
Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices
with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$.
All possible inertia values, ranks, and multiple eigenvalues
of $A + B$ are determined. Extension of the results to the sum of $k$ matrices
with $k > 2$ and connections of the results to other subjects such
as algebraic combinatorics are also discussed.
Keywords:complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues Categories:15A42, 15A57 |
2. CJM 2003 (vol 55 pp. 91)
Some Convexity Features Associated with Unitary Orbits Let $\mathcal{H}_n$ be the real linear space of $n\times n$ complex
Hermitian matrices. The unitary (similarity) orbit $\mathcal{U}
(C)$ of $C \in \mathcal{H}_n$ is the collection of all matrices
unitarily similar to $C$. We characterize those $C \in \mathcal{H}_n$
such that every matrix in the convex hull of $\mathcal{U}(C)$ can
be written as the average of two matrices in $\mathcal{U}(C)$. The
result is used to study spectral properties of submatrices of
matrices in $\mathcal{U}(C)$, the convexity of images of $\mathcal{U}
(C)$ under linear transformations, and some related questions
concerning the joint $C$-numerical range of Hermitian matrices.
Analogous results on real symmetric matrices are also discussed.
Keywords:Hermitian matrix, unitary orbit, eigenvalue, joint numerical range Categories:15A60, 15A42 |
3. CJM 2002 (vol 54 pp. 571)
Diagonals and Partial Diagonals of Sum of Matrices Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a
certain group action such as
\begin{enumerate}[(4)]
\item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices
$A$ by $(U,V)*A = UAV^t$,
\item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m
\times n$ real matrices $A$ by $(U,V)*A = UAV^t$,
\item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or
skew-symmetric matrices $A$ by $U*A = UAU^t$,
\item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or
skew-symmetric matrices $A$ by $U*A = UAU^t$.
\end{enumerate}
Denote by
$$
\mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in
\mathcal{O}(A_i), i = 1,\dots,k\}
$$
the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of
diagonals 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 lie
in 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 many
cases, complete description of these sets is given in terms of the
inequalities involving the singular values of $A_1,\dots,A_k$. 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 $\mathcal{O}(A_1,\dots,A_k)$ of complex
Hermitian matrices under the action of unitary similarities are
also discussed.
Keywords:orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular values Categories:15A42, 15A18 |