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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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    Is `h^{*}` something like the Hermitian conjugate? And when you say 'Dirac delta function', do you really mean the Kronecker delta (i.e. 1 when k is zero, 0 otherwise)? Otherwise, I'm not sure how to make sense of this.2012-01-13
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    Also, are you expecting to find a value of h which satisfies this for all k, or just one given value of k? If the former, I imagine there are only solutions in certain special cases, if at all. In the latter case, the solutions won't be unique, will they?2012-01-13
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    Yes. h^{*} is the Hermitian conjugate and Dirac delta "function" is \delta_{k}=1 when k=0 and 0 otherwise. I expect to find a solution h satisfying all these equations for all k being non-negative (even negative if \Phi is full rank). The solution is not unique in general. That's the reason why I'm searching for a formula for h (maybe a subclass of all the solutions. That would be great too). I guess it's hard to find an explicit formula for h. Are there any special properties for h? any clues? Thanks2012-01-13
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    That isn't what is normally meant by the [Dirac delta function](http://en.wikipedia.org/wiki/Dirac_delta_function). What you are describing is (at least in my experience) usually called the [Kronecker delta](http://en.wikipedia.org/wiki/Kronecker_delta).2012-01-13
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    I suspect that you would have much more luck with this problem over at math.stackexchange.2012-01-13
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    Sorry, I'm still not sure I am fully understanding this: there must be a large class of diagonalizable matrices Phi for which there are no solutions for h, for example any identity matrix.2012-01-13
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    Sorry, my fault. \delta_{k} is the Kronecker delta. I don't know what exactly \Phi is either. The only thing I know is \Phi is diagonalizable. I'm also interested in what properties \Phi should have to ensure the existence of h. Essentially, I just hope to find some clues and pointers on similar problems. I guess this problem has been studied but I don't know the name and how to find them. By the way, is there any suggestions on how to search math problems on google, since I got rubbish by searching on google? Thanks2012-01-13
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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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