Noncommutative Rings

Abstract

The set of linear transformations of a finite-dimensional vector space has two operations defined on it, addition and multiplication; writing out linear transformations in terms of matrixes, these operations can be transferred to matrixes as well. The existence of both these operations is extremely important and is constantly used. It is, for example, only because of this that we can define polynomials in a linear operator; and, among other uses, they are used in the study of the structure of a linear transformation, which depends in an essential way on the multiplicity of roots of its minimal polynomial. The same two operations, together with a passage to limits, make it possible to define analytic functions of a (real or complex) matrix.