On the matrix equation f(X)=A

Abstract

This paper deals with the matrix equation f(X)=A, where A∈Cn×n is a given matrix, and ƒ is a complex holomorphic function defined on an open subset Dƒ of C. It furnishes information concerning the set S of all solutions X∈Cn×n of ƒ(X)=A, the subset Sp ⊂ S of all solutions that are polynomials in A, and the subset SD ⊂ S of all diagonalizable solutions. A necessary and sufficient condition for S to be nonempty is established. Relations among all solutions of ƒ(X)=A lying in the same similarity class are established. In the particular case where A∈Rn×n, Dƒ is symmetric with respect to the real line R ⊂ C, and ƒ(z̄)=f(z) for every z∈Dƒ, a necessary and sufficient condition for S ∩ Rn×n to be nonempty is established. Necessary and sufficient conditions for SP and SD to be empty, finite, infinite, and commutative are established. Explicit computations of all the elements of SP and SD are furnished, and the numbers of elements of these two sets are determined. Necessary and sufficient conditions for the relations S=SP, S=SD, or SP=SD to be satisfied are established. Necessary and sufficient conditions for the existence of normal, hermitian, skew-hermitian, positive definite, and unitary solutions are established. It is shown that the set P(ƒ,A) of all polynomials p ∈ C[x] such that X = p(A) is a solution of ƒ(X) = A depends only on the minimal polynomial of A, and not on the size of A. An example is furnished of a matrix equation ƒ(X) = A that has no solution in Fn×n for any field F, but has a solution in Rn×n when R is a certain ring. These results are applied to the particular equations Xm = A, eX = A and the fixed-point equation ƒ(X) = X.