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.