Some computational problems and methods related to invariant factors and control theory

Abstract

This chapter discusses computational problems and methods related to invariant factors and control theory. The determination of many invariants in linear algebra, for instance, minimal polynomials of a vector or a matrix, invariant subspaces, the rational canonical form of a matrix, the number of eigenvalues of a matrix in a half-plane or a circle, requires computation in the polynomial rings R[z] or C[z]. These computations are rather awkward because they involve checks of divisibility that must be exact and polynomial arithmetic (especially matrix-valued polynomial arithmetic) is very awkward to program. As the numbers desired are often integers (the degree of the minimal polynomial of a matrix), these problems tend to have some of the flavor of finite algebra, even though strictly speaking they belong to linear algebra. It is well known that the common factor of two polynomials can be determined by the Euclidean algorithm.