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.