Let $A \in \mathbb{R}^{m \times m}$ be a nonsymmetric zero diagonal matrix with a zero/non-zero pattern which is symmetric and persymmetric (i.e. symmetric in the northeast-to-southwest diagonal).
If $A$ has any kind of block checkerboard pattern, or an offset of a block checkerboard pattern that conserves symmetry and persymmetry (of the pattern), then $A^k, k=2n+1, n\in\mathbb{N}_0$ has zero diagonal.
1) Is there a necessary and sufficient condition on the zero/non-zero pattern to get a zero diagonal $A^k, k=2n+1, n\in\mathbb{N}_0$?
If $A^k, k=2n+1, n\in\mathbb{N}_0$ is traceless, then the spectrum of $A$ is symmetric with respect to the imaginary axis.
2) Is the numerical range (i.e. field of values, $W(A)=\left\{ \frac{v^* A v}{v^* v}, v \in \mathbb{C}^m, v\ne 0 \right\}$) always symmetric with respect to the imaginary axis as well?
Examples: \begin{align} \pmatrix{0 &2 &0 &-4\\ 1 &0 &2 &0 \\ 0 &-1 &0 &8 \\ -4 &0 &7 &0} \end{align}
\begin{align} \pmatrix{0 &0 &3 &-4\\ 0 &0 &-1 &3 \\ 3 &-1 &0 &0 \\ -4 &3 &0 &0} \end{align}
\begin{align} \pmatrix{0 &-9 &1 &0\\ -9 &0 &0 &-1 \\ -1 &0 &0 &1 \\ 0 &5 &7 &0} \end{align}
