91Ó°ÊÓ

An eigenvalue ofAis a scalar $\mathbf{λ}$such that the equation$\mathbf{Av}\mathbf{=}\mathbf{λ\pm ¹}$has a nontrivial solution.

Assume that, A is an invertible matrix of order $n×n$, and $\stackrel{â‡\P Ä}{\mathrm{v}}$ is an eigenvector of A corresponding to eigenvalue $\mathrm{λ}$.

Is role="math" localid="1659532740882" $\stackrel{â‡\P Ä}{\mathrm{v}}$an eigenvector of $A^3$or not, and find the eigenvalue of $A^3$.

If $\stackrel{â‡\P Ä}{\mathrm{v}}$ is an eigenvector of matrix A then,

$A\stackrel{â‡\P Ä}{\mathrm{v}}=\mathrm{λ}\stackrel{â‡\P Ä}{\mathrm{v}}$

By the properties of eigenvalues and eigenvectors, if $\stackrel{â‡\P Ä}{\mathrm{v}}$is an eigenvector of matrix A, then $\stackrel{â‡\P Ä}{\mathrm{v}}$ is an eigenvector of matrices$A^2,A^3,....,A^n$ , as shown below.

$$\begin{gathered} A^2\stackrel{â‡\P Ä}{\mathrm{v}}={\mathrm{λ}}^2\stackrel{â‡\P Ä}{\mathrm{v}} \\ {A}^3\stackrel{â‡\P Ä}{\mathrm{v}}={\mathrm{λ}}^3\stackrel{â‡\P Ä}{\mathrm{v}} \\ {A}^n\stackrel{â‡\P Ä}{\mathrm{v}}={\mathrm{λ}}^n\stackrel{â‡\P Ä}{\mathrm{v}} \end{gathered}$$

Where, n is positive integer.

Now look at$A^3$obtained as,

Thus,$\stackrel{â‡\P Ä}{\mathrm{v}}$is an eigenvector of$A^3$corresponding to eigenvalue${\mathrm{λ}}^3$ .

Hence, $\stackrel{â‡\P Ä}{\mathrm{v}}$is an eigenvector of$A^3$ , and the corresponding eigenvalue is${\mathrm{λ}}^3$ .