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Thread: Generalized Eigenspaces

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    Generalized Eigenspaces

    Let $\displaystyle V = W_1 \oplus W_2, \phi: V \rightarrow V $a linear map. Prove that $\displaystyle V(\lambda)^+ = W_1(\lambda)^+ \oplus W_2(\lambda)^+.

    $ Injectivity is easy enough, but I can't seem to prove $\displaystyle V(\lambda)^+ = W_1(\lambda)^+ + W_2(\lambda)^+ $ without assuming that $\displaystyle W_1, W_2 $ are $\displaystyle \phi-$stable. Am I missing something here, or is this assumption required for the implication to hold?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by h2osprey View Post
    Let $\displaystyle V = W_1 \oplus W_2, \phi: V \rightarrow V $a linear map. Prove that $\displaystyle V(\lambda)^+ = W_1(\lambda)^+ \oplus W_2(\lambda)^+.

    $ Injectivity is easy enough, but I can't seem to prove $\displaystyle V(\lambda)^+ = W_1(\lambda)^+ + W_2(\lambda)^+ $ without assuming that $\displaystyle W_1, W_2 $ are $\displaystyle \phi-$stable. Am I missing something here, or is this assumption required for the implication to hold?
    You need to define some things here friend. Presumably $\displaystyle S(\lambda)$ is the eigenspace of the eigenvalue $\displaystyle \lambda$? What does the $\displaystyle ^+$ mean?
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    Whoops my bad. $\displaystyle V(\lambda)^+$ is defined as the generalized eigenspace of the eigenvalue $\displaystyle \lambda$, i.e. $\displaystyle V(\lambda)^+ := \{v \in V | (\phi - \lambda .Id)^n (v)= 0 $ for some $\displaystyle n \in \mathbb{N}\}$
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