Computation of roots of real and complex matrices (Q1067358)

A method is presented for the computation of the r-th root of an arbitrary real matrix A which does not use the eigenvalues and eigenvectors of A. The basic tool is a spectral decomposition of A which is computed with the aid of the sign function of A. The decomposition has as one factor a matrix \(A_ 1\) with all eigenvalues in the right half plane and it remains to compute the r-th root of \(A_ 1\). This is done with the Hoskins-Walton method, which is essentially the Newton method for \(f(X)=X^ r-A_ 1\). The method is generalized to work for \(n\times n\) complex matrices by representing them in the usual way as real matrices of dimension 2n. Some example are given.