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Could someone give me some hint about a possible method to find the function $f$ which solve this equation: $$f(H^2)=\alpha f(H)$$ where $\alpha$ a constant with $\alpha \in C$ and: $$H=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$ a matrix with complex elements?

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$f$ give a matrix or a complex number? –  Davide Giraudo Jun 28 '12 at 12:01
$f$ should give a matrix –  Riccardo.Alestra Jun 28 '12 at 12:04
You certainly need to put some conditions on $f$, unless you are happy with the solution $f=0$. –  Martin Argerami Jun 29 '12 at 17:07
@Martin Argerami: I agree. Obviously the trivial solution $f=0$ has to be rejected. –  Riccardo.Alestra Jul 2 '12 at 7:12

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