Question

Suppose that λ is an eigenvalue of A. Show that λ2 is then an eigenvalue of A2

Answer 1

Answer with Step-by-step explanation:

We are given that $\lambda$ is an eigenvalue of A.

We have to prove that $\lambda^2$ is an eigenvalue of $A^2$

$\lambda$ is eigenvalue of A then, there exist an eigen vector x such that

$Ax=\lambda x$

Multiply by A on both sides then we get

$A(Ax)=A(\lambda x)$

$A^2x=\lambda(Ax)$

$A^2x=\lambda(\lambda x)$

$A^2x=\lambda^2x$

Therefore, $\lambda^2$ is an eigen value of $A^2$

Hence, proved.