December 27, 2004
Leiba Rodman
Title: Well Posedness in Matrix Analysis
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 , and for each , a set is given. An element will be called well posed (relative to the sets and ) if for every close enough to there is a that is as close to 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 is the set of all -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.