### Pal Rozsa, Technical University of Budapest

A wide range of problems occuring in science and engineering lead to
systems of linear algebraic or differential equations with coefficient
matrices of special structure. Theis series of three lectures gives
a survey on their properties and on some of their applications.

**Lecture 1: ***Block Matrices*, Tuesday, January 5, 1993

Matrices partitioned into four blocks can be considered as the simplest
block matrices. Their factorization leads to some results concerning
the inverse of a sub matrix and the inverse of modified matrices. In
particular, conditions are given for a certain eigenvalue of the fiven
matrix to be invariant with respect of the modification; the change
of the corresponding eigenvector can be determined as well.

The problem of finding the spectral decomposition of block matrices
with commutative blocks can be reduced to the eigenvalue problem of
matrices of lower order; the Kronecker products of their eigenvectors
yield the eigenvectors of the given block matrix. In the special case
of Kronecker polynomials the results can be applied for solving certain
matrix equations and systems of algebraic or differential equations
as well.

**Lecture 2: ***Generalized Band Matrices and Their Inverses*,
Tuesday, January 12, 1993

The inverse of a non-singular tridiagonal matrix is a one-pair-matrix,
i.e. the rank of any sub matrix in its upper and lower tridiagonal parts
is one. That means, subtracting appropriate one-rank-matrices from the
upper and lower triangular part of the inverse we get zero off-diagonal
elements and the main diagonal forms an "overlapping" area. The same
idea can be applied for block tridiagonal matrices with non-singular
blocks in the codiagonals. Considering the special case of band matrices,
interesting structural properties can be found for their inverses, called
"semiseparable" matrices. A possible generalization is based on the
fact that both the band matrices and their inverses have low-rank submatrices
in their upper right and lower left corners. The generalized band matrices
make it possible to characterize a wide range or matrices, among other
sparse matrices.

**Lecture 3: ***Application of Block Matrices*, Tusday, January
19, 1993

The vibrations of certain corpuscular systems in two and three dimensions
can be described by using Kronecker polynomials. Therefore the spectral
decomposition of Kronecker polynomials can be applied for solving the
corresponding systems of differential equations. - Certain matrix equations
occurring in control theory, e.g. the Lyapunov equation and related
systems of differential equations can be solved by using Kronecker polynomials.
- Both the ordinary and partial differential equations can be discredited
by making use of the finite differences; the obtained difference equations
can be written as systems of algebraic equations with block tridiagonal
or block band matrices as coefficients. Properties of their inverses
- often called Green matrices - can be characterized by applying the
results on generalized band matrices. - If the system has a periodic
structure, i.e. it can be characterized by a periodic tridiagonal matrix,
the corresponding structure, i.e. it can be characterized by a periodic
tridiagional matrix, the corresponding characteristic polynomial can
be factorized in certain cases. This interesting phenomenon can be generalized
for periodic block tridiagonal matrices under very special conditions
only.