Question

If V is an eigenvector of an n*n invertible matrix A, then V is also an eigenvector of the matrix A2.

Answer 1

Answer on Question #40381, Math, Linear Algebra

If $V$ is an eigenvector of an $n \times n$ invertible matrix $A$, then $V$ is also an eigenvector of the matrix $A^2$.

Solution.

Suppose

$$A \boldsymbol {V} = \lambda \boldsymbol {V}$$

Then we have

$$A ^ {2} \boldsymbol {V} = A (A \boldsymbol {V}) = A (\lambda \boldsymbol {V}) = \lambda A \boldsymbol {V} = \lambda^ {2} \boldsymbol {V}.$$

Answer:

So $\pmb{V}$ is an eigenvalue of $A^2$ with eigenvalue $\lambda^2$.