Normal Matrices Over an Arbitrary Field of Characteristic Zero

Abstract

Introduction. A matrix A with elements in the complex number field is said to be a normal matrix, if AA* = A*A, where A* is the conjugate transposed of the matrix A. A necessary and sufficient condition that A be a normal matrix is that there exist a unitary matrix U, such that U*AUD, where D is a diagonal matrix. Further a matrix A is normal, if, and only if, A*f(A), where f(x) is a polynomial in x.1 From this it is possible to make a satisfactory definition of normality with respect to any non-singular hermitian matrix H. A matrix A is said to be normal with respect to the hermitian matrix H, if AH =Hf (A*). The simplicity of the canonical form of a normal matrix A under unitary transformation suggests the following problems. What are the possible canonical forms for a matrix A, normal with respect to H, under similarity transformations by matrices which are conjunctive automorphs of H? What are necessary and sufficient conditions that two matrices, both normal with respect to H, be similar under transformations by matrices, which are conjunctive automorphs of H. These problems were discussed in a previous paper.' It is our intention here to consider the corresponding problems where the matrices under consideration are matrices over a field of characteristic zero.