Generalized Eigenspaces

**h2osprey** · April 1st 2011, 10:55 AM

Let $V = W_1 \oplus W_2, \phi: V \rightarrow V$ a linear map. Prove that $V(\lambda)^+ = W_1(\lambda)^+ \oplus W_2(\lambda)^+.<br /> <br />$ Injectivity is easy enough, but I can't seem to prove $V(\lambda)^+ = W_1(\lambda)^+ + W_2(\lambda)^+$ without assuming that $W_1, W_2$ are $\phi-$ stable. Am I missing something here, or is this assumption required for the implication to hold?

**Drexel28** · April 1st 2011, 01:08 PM

Originally Posted by h2osprey

Let $V = W_1 \oplus W_2, \phi: V \rightarrow V$ a linear map. Prove that $V(\lambda)^+ = W_1(\lambda)^+ \oplus W_2(\lambda)^+.<br /> <br />$ Injectivity is easy enough, but I can't seem to prove $V(\lambda)^+ = W_1(\lambda)^+ + W_2(\lambda)^+$ without assuming that $W_1, W_2$ are $\phi-$ 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 $S(\lambda)$ is the eigenspace of the eigenvalue $\lambda$ ? What does the $^+$ mean?

**h2osprey** · April 1st 2011, 01:44 PM

Whoops my bad. $V(\lambda)^+$ is defined as the generalized eigenspace of the eigenvalue $\lambda$ , i.e. $V(\lambda)^+ := \{v \in V | (\phi - \lambda .Id)^n (v)= 0$ for some $n \in \mathbb{N}\}$

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