I'm trying to find the general expression for $h$, an $n \times 1$ vector, which solves
$h^{T} \Phi^{2k} h = \delta_{k} $
where $k$ is a non-negative integer, $\Phi$ is an $n \times n$ diagonalizable matrix (assume $\Phi$ is full rank) and $\delta$ is the Kronecker delta function $ \delta_k = \left\{\begin{array}{ccc} 1 & & k = 0 \\ 0 & & k \neq 0 \end{array}\right. $
Is there any specific name for such problem? I appreciate if someone could provide me with some pointers.