Block eigenvalue decomposition using nth roots of the identity matrix

The matrix sign function has been utilized in recent years for block diagonalization of complex matrices. In this paper, nth roots of the identity matrix including the matrix sector function are utilized for block diagonalization of general matrices. Specifically, we derive classes of rational fixed point functions for nth roots of any nonsingular matrix which are then used for block eigen-decomposition. Based on these functions, algorithms may have any desired order of convergence are developed. Efficient implementation of these algorithms using the QR factorization is also presented. Several examples are presented to illustrate the performance of these methods.