Eigenvectors of the square of the matrix

Question

Given $A: n\times n$ matrix with eigenvector $w$ for eigenvalue $c$, does $B$, where $B^2 = A$ have $w$ as an eigenvector?

I.e, $A*w = B*B*w = c*w$. Is $w$ an eigenvector of $b$ with eigenvalue $\sqrt{c}$? I know that $A^2*w = A*A*w = A*c*w = c^2*w$ implies $A^2$ has eigenvector $w$.

Answer 1

No. An example is the real square matrix $$B= \left[ \begin{matrix}0 &1 \\ -1 & 0 \end{matrix} \right]$$

which has no real eigenvalues (its characteristic polynomial is $x^2+1$ which has no real roots). However, $B^2 = -1$, so it has $-1$ as an eigenvalue, and all vectors are eigenvectors.

Answer 2

Crostul's answer is correct but I would like to give some intuition about how this situation is possible. If a matrix $B$ rotates vectors by a right angle then it has no eigenvectors. However when this matrix is applied twice, it inverts all vectors so $-1$ is an eigenvalue.