Differential Equations and Linear Algebra, 4th Edition

6.3 Applications Involving Powers of Matrices

In this section we discuss applications that depend on an ability to compute the matrix ${\mathbf{\text{A}}}^k$ for large values of k, given the $n\times n$ matrix A. If A is diagonalizable, then ${\mathbf{\text{A}}}^k$ can be found directly by a method that avoids the labor of calculating the powers ${\mathbf{\text{A}}}^2,{\mathbf{\text{A}}}^3,{\mathbf{\text{A}}}^4,\dots$ by successive matrix multiplications.

Recall from Section 6.2 that, if the $n\times n$ matrix A has n linearly independent eigenvectors ${\mathbf{\text{v}}}_1,{\mathbf{\text{v}}}_2,\dots ,{\mathbf{\text{v}}}_n$ associated with the eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ (not necessarily distinct), then

$$\mathbf{\text{A}}=\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1},$$ (1)

where

$$\mathbf{\text{P}}=\left[\begin{array}{ccc}|& |& |\\ {\mathbf{\text{v}}}_1& {\mathbf{\text{v}}}_2& \cdots & {\mathbf{\text{v}}}_n\\ |& |& |\end{array}\right]\text{and}\mathbf{\text{D}}=\left[\begin{array}{cccc}{\lambda }_1& 0& \cdots & 0\\ 0& {\lambda }_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\lambda }_n\end{array}\right].$$

Note that (1) yields

$${\mathbf{\text{A}}}^2=(\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1})(\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1})=\mathbf{\text{P}}\mathbf{\text{D}}({\mathbf{\text{P}}}^{-1}\mathbf{\text{P}})\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1}=\mathbf{\text{P}}{\mathbf{\text{D}}}^2{\mathbf{\text{P}}}^{-1}$$

because ${\mathbf{\text{P}}}^{-1}\mathbf{\text{P}}=\mathbf{\text{I}}$. More generally, for each positive integer k,

$$\begin{array}{rcl}{\mathbf{\text{A}}}^k& =& {(\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1})}^k\\ =& (\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1})(\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1})\cdots (\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1})(\mathbf{\text{P}}\mathbf{\text{D}}{\mathbf{\text{P}}}^{-1})\end{array}$$