Mth Power of an N × N Matrix and Its Connection with the Generalized Lucas Polynomials

Abstract

The Mth power of an N × N matrix is expressed via the Cayley‐Hamilton theorem as a linear combination of the lower powers of the matrix. The polynomial coefficients of the lower powers of the matrix are expressed in terms of polynomials in N variables, termed the generalized Lucas polynomials. The independent variables in the generalized Lucas polynomials are the traces of the lower powers of the matrix.