In this section we discuss applications that depend on an ability to compute the matrix for large values of k, given the matrix A. If A is diagonalizable, then can be found directly by a method that avoids the labor of calculating the powers by successive matrix multiplications.
Recall from Section 6.2 that, if the matrix A has n linearly independent eigenvectors associated with the eigenvalues (not necessarily distinct), then
Note that (1) yields
because . More generally, for each positive integer k,