An example of a $2 \times 2$ matrix $A$ without real eigenvalues and s.t $A^2$ has $-1$ as an eigenvalue with algebraic and geometric multiplicity $2$

Basically I tried the matrix \begin{bmatrix} 0 & -1\\ 1 & 0\\ \end{bmatrix}

but this eigenvalue of $A^2$ is $1$,and $-1$ which has algebraic multiplicity is $1$. I can not find any $2 \times 2$ matrix that satisfies both has $-1$ as an eigenvalue and algebraic multiplicity $2$.

1 Answer

Well basically the matrix $$ \begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix} $$ works. The eigenvalue of this matrix is $\lambda^2 + 1$ and the eigenvalue of this matrix $A^2$ is $(\lambda +1)(\lambda + 1)$ where both algebraic and geometric multiplicity 2