On quasi-commutative matrices

Abstract

where c is a scalar matrix. It is well knownf that a relation of this type can not be satisfied by finite matrices. However, in the calculation of commutation formulas for polynomials in p and q no use is made of the fact that c is a scalar but merely that it is commutative with both p and q.% And there do exist pairs of finite matrices x, y of the same order such that xy—yx is not zero and is commutative with both x and y. Such matrices will be called quasi-commutative matrices and either may be said to be quasi-commutative with the other. In a certain sense the algebra of polynomials in a pair of quasi-commutative matrices is homeomorphic to the algebra arising in quantum mechanics. It is hoped to discuss such algebras in some detail in a later paper. In the present paper we shall make a brief study of quasi-commutative matrices whose elements belong to the complex number field. The concept of quasi-commutativity is an extension or generalization of commutativity, and as would be expected, some of the results obtained are generalizations of known theorems concerning commutative matrices. The problem of determining quasi-commutative matrices is that of finding matrices x, y, z (^0) which satisfy the equations