On the matrix roots of \(f(X)=A\)
Let A be an \(n\times n\) matrix with complex elements and let f(x) be a non-constant polynomial with complex coefficients. The author considers the problem of finding an \(n\times n\) matrix B with \(f(B)=A\). He shows, for example, that if A is the Jordan block with eigenvalue \(\lambda\), such a B exists if and only if either \(n=1\) or there exists a complex number \(\gamma\) such that \(f(\gamma)=\lambda\) and \(f'(\gamma)\neq 0\). In the case that A is an \(n\times n\) matrix with a single eigenvalue he gives necessary and sufficient conditions for the existence of an irreducible solution X to \(f(X)=A\). He also shows that if A is a nonderogatory matrix and I is the ideal generated by the characteristic polynomial of A then there exists a matrix B with \(f(B)=A\) if and only if there exists a polynomial g(x) with \(f(g(x))\equiv x(mod I)\).
scientific article; zbMATH DE number 4071108