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1. CJM 2004 (vol 56 pp. 776)
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices We explicitly describe
the best approximation in
geodesic submanifolds of positive definite matrices
obtained from involutive
congruence transformations on the
Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of
positive definite matrices.
An explicit calculation for the minimal distance
function from the geodesic submanifold
${\mathrm{Sym}}(p,{\mathbb R})^{++}\times
{\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in
${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is
given in terms of metric and
spectral geometric means, Cayley transform, and Schur
complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$
Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform Categories:15A48, 49R50, 15A18, 53C3 |
2. CJM 2003 (vol 55 pp. 1000)
Some Convexity Results for the Cartan Decomposition In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$
where $a(g)$ is the abelian part in the Cartan decomposition of
$g$. This is exactly the support of the measure intervening in the
product formula for the spherical functions on symmetric spaces of
noncompact type. We give a simple description of that support in
the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$,
$\mathbf{C}$ or $\mathbf{H}$. In particular, we show that
$\mathcal{S}$ is convex.
We also give an application of our result to the description of
singular values of a product of two arbitrary matrices with
prescribed singular values.
Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values Categories:43A90, 53C35, 15A18 |
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 |
4. CJM 2001 (vol 53 pp. 758)
Inequivalent Transitive Factorizations into Transpositions The question of counting minimal factorizations of permutations into
transpositions that act transitively on a set has been studied extensively
in the geometrical setting of ramified coverings of the sphere and in the
algebraic setting of symmetric functions.
It is natural, however, from a combinatorial point of view to ask how such
results are affected by counting up to equivalence of factorizations, where
two factorizations are equivalent if they differ only by the interchange of
adjacent factors that commute. We obtain an explicit and elegant result for
the number of such factorizations of permutations with precisely two
factors. The approach used is a combinatorial one that rests on two
constructions.
We believe that this approach, and the combinatorial primitives that have
been developed for the ``cut and join'' analysis, will also assist with the
general case.
Keywords:transitive, transposition, factorization, commutation, cut-and-join Categories:05C38, 15A15, 05A15, 15A18 |