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Here is a problem that came up in my research:

Let $X,U$ be finite dimensional vector spaces over some field $F$ and let $f: X \rightarrow X, \beta : U \rightarrow X, \kappa : X \rightarrow U$ be linear maps. Then $(f+\beta \circ \kappa)$ is an endomorphism of $X$. Let $X = \oplus_{i=1}^r X_i$ be the decomposition (possibly trivial) of $X$ into $(f+\beta \circ \kappa)$-invariant subspaces.

The problem statement is the following: given $f, \beta$ as above, under what conditions does some $\kappa$ as above exist, such that i) $X$ admits non-trivial decomposition $X=\oplus_{i=1}^r X_i$ into $(f+\beta \circ \kappa)$-subspaces and ii) $\pi_i \circ \beta(U) = \beta(U) \cap X_i$, where $\pi_i : X \rightarrow X_i$ is the projection of $X$ onto $X_i$ and $\beta (U)$ is the image of $U$ under $\beta$ or equivalently $\beta(U) = \oplus_{i=1}^r \left[\beta(U) \cap X_i \right]$?

It seems to me that this is a possibly intractable problem, since i don't have any idea from where to start. Any particular branch of algebra that could potentially be used to make a start? What is your opinion?

thanks :-)

Clarification: Given finite dimensional vector space $X$ and $f \in End(X)$, there always exists decomposition of $X$ into $f$-invariant subspaces. The decomposition corresponds to the rational canonical form of $f$ and it might be trivial, i.e. $i=1$ and $X_i=X$. In my problem i want the decomposition to be non-trivial.

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Is there a mistake in your question? You have no conditions on $\kappa$ and do you asume that some decomposition of $X$ is given? –  Simon Markett May 1 '12 at 16:28
    
Thanks, i fixed this. –  Manos May 1 '12 at 16:34

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