6.3 Applications Involving Powers of Matrices

In this section we discuss applications that depend on an ability to compute the matrix Ak for large values of k, given the n×n matrix A. If A is diagonalizable, then Ak can be found directly by a method that avoids the labor of calculating the powers A2,A3,A4, by successive matrix multiplications.

Recall from Section 6.2 that, if the n×n matrix A has n linearly independent eigenvectors v1,v2,,vn associated with the eigenvalues λ1,λ2,,λn (not necessarily distinct), then

A=PDP1, (1)



Note that (1) yields


because P1P=I. More generally, for each positive integer k,


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