Matrix Problems

Abstract

In this chapter certain classification problems are solved. In each case an equivalence relation on the set of all matrices or the set of all matrices with a given subdivision is given and a “normal form” is sought: that is a family of matrices such that of each equivalence class one and only matrix, the representative, is chosen.

There is little theory within this chapter as the main goal is to establish the normal form of certain examples: the first ones are taken from linear algebra, then some more complicated are tackled. All of them are important in the theory and will be used frequently in the book for illustration. The phenomena which can be observed is that there are three distinct cases of normal forms: either it is a finite list, or it is an infinite but complete list, where for each size of matrices almost all members of the normal form are parametrized by one variable, or there exists one size of matrices for which there are representatives which are parametrized by two variables.

The technique applied to obtain the normal form is simple reduction using the allowed row and column transformations of the matrices. It is a powerful machinery and gives a fast start. But it also has its drawbacks: first it is quite easy to oversee errors and the technique delivers just a list without any internal structure.