Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2011 (vol 64 pp. 805)
Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process Categories:46L53, 60B20, 14L24 |
2. 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 |
3. CJM 2009 (vol 61 pp. 351)
Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicity-free. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual Littlewood--Richardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials Categories:14L30, 22E46 |
4. CJM 2008 (vol 60 pp. 1050)
Adjacency Preserving Maps on Hermitian Matrices Hua's fundamental theorem of the geometry of hermitian matrices
characterizes bijective maps on the space of all $n\times n$
hermitian matrices preserving adjacency in both directions.
The problem of possible improvements
has been open for a while. There are three natural problems here.
Do we need the bijectivity assumption? Can we replace the
assumption of preserving adjacency in both directions by the
weaker assumption of preserving adjacency in one direction only?
Can we obtain such a characterization for maps acting between the
spaces of hermitian matrices of different sizes? We answer all
three questions for the complex hermitian matrices, thus obtaining
the optimal structural result for adjacency preserving maps on
hermitian matrices over the complex field.
Keywords:rank, adjacency preserving map, hermitian matrix, geometry of matrices Categories:15A03, 15A04, 15A57, 15A99 |
5. CJM 2005 (vol 57 pp. 871)
Hermitian Yang-_Mills--Higgs Metrics on\\Complete KÃ¤hler Manifolds In this paper, first, we will investigate the
Dirichlet problem for one type of vortex equation, which
generalizes the well-known Hermitian Einstein equation. Secondly,
we will give existence results for solutions of these vortex
equations over various complete noncompact K\"ahler manifolds.
Keywords:vortex equation, Hermitian Yang--Mills--Higgs metric,, holomorphic vector bundle, KÃ¤hler manifolds Categories:58E15, 53C07 |
6. 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 |
7. 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 |