The nth power of a matrix and approximations for large n

Abstract. When a square matrix A, is diagonalizable, (for example, when A is Hermitian or has distinct eigenvalues), then An can be written as a sum of the nth powers of its eigenvalues with matrix weights. However, if a 1 occurs in its Jordan form, then the form is more complicated: An can be written as a sum of polynomials of degree n in its eigenvalues with coefficients depending on n. In this case to a first approximation for large n, An is proportional to nm−1λn with a constant matrix multiplier, where λ is the eigenvalue of maximum modulus and m is the maximum multiplicity of λ