December 27, 2004

Leiba Rodman

Title: Well Posedness in Matrix Analysis

Abstract:

A review of several results will be presented and open problems stated, concerning description of all well posed solutions to various matricial problems in the following sense:

Let be given a class of real or complex matrices $ \mathcal A$, and for each $ X\in \mathcal A$, a set $ \mathcal G(X)$ is given. An element $ Y_0 \in \mathcal G(X_0)$ will be called well posed (relative to the sets $ \mathcal A$ and $ \mathcal G(X)$) if for every $ X\in \mathcal A$ close enough to $ X_0$ there is a $ Y \in \mathcal G(X)$ that is as close to $ Y_0$ as we wish.

The following topics are planned to be covered, time permitting:

1. Matrix decompositions: polar decompositions with respect to an indefinite inner product, Cholesky factorizations, singular value decomposition.

2. Invariant subspaces of matrices; here the set $ \mathcal G(X)$ is the set of all $ X$-invariant subspaces.

3. Invariant subspaces of matrices with symmetries related to indefinite inner products. The invariant subspaces in question include semidefinite and neutral subspaces (with respect to an indefinite inner product).

4. Applications of invariant subspaces of matrices with or without symmetries. The applications include: general matrix quadratic equations, the continuous and discrete algebraic Riccati equations, minimal factorization of rational matrix functions with symmetries and the transport equation from mathematical physics.

The talk is based on a review paper with A. C. M. Ran.