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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.

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    thanks for the improvement. It looks better. I'm trying to see whether I could get some clues by searching the column space and row space. Sounds like the projection onto the row space of $\Phi^{k}$ is orthogonal to that onto column space. Cheers!2012-01-15

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