Operator Matrices

Abstract

The assertion of Problem 70 is often useful in operator theory, but, it turns out, the context it properly belongs to is a much more general part of algebra. If an operator is represented as a matrix whose entries are commutative operators, then it is profitable to consider the (commutative) ring with unit generated by those entries. Commutative rings have a determinant theory that is not much more frightening than in the numerical case. Indeed, if S is a finite square matrix over a commutative ring M, then det S makes sense, as an element of M: just apply the usual definition, according to which the determinant of a matrix of size n is a sum of n! terms with appropriate signs.