MATLAB: Diagonalization

In this activity you will find the matrix $P$ that diagonalizes a given matrix and the resulting diagonal matrix $D,$ and utilize these to

indirectly find a higher power of the given matrix.

Consider the matrix $A.$

Higher powers of A can be found indirectly using the diagonalization matrix $P$ and resulting diagonal matrix $D.$ These matrices were

found using the eig$()$ command in zyLab $6\mathrm{.}2.$ The matrix $A$ to the kth power can be indirectly calculated by $A^k=P{D}^k{P}^{-1}.$

$A=[11$;$-24]$

$[P,D]=$eig$(A)$

$\%$Indirectly find $A^5$ using $P$ and $D.$ The command power$(\{$:$D,5)$ raises every element in the diagonal matrix $D$

$\%$to the fifth power.

A$5$indirectly $=P^{**}$ power $(D,5)**$ inv $(P)$

$\%$The same result can be found directly, but the direct method is not computationally practical for higher powers.

A$5$directly $=A^5$

Use the following matrix for this activity.

$C=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$