91Ó°ÊÓ

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

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

Is $\stackrel{â‡\P Ä}{v}$an eigenvector of $A^3$or not, and find the eigenvalue of $A^3$.

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

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

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

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

Where, n is positive integer.

Now look at $A^3$obtained as,

$A^3\stackrel{â‡\P Ä}{v}=\mathrm{λ}\stackrel{â‡\P Ä}{v}$

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

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