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Let V be a finite dimensional vector space over $F$, let $S,T:V\rightarrow V$ be linear operators on $V$, and assume that $S$ is invertible. Let $\lambda \in F$ be an eigenvalue of T, and let $V_\lambda$ be the corresponding eigenspace.

a) Prove that $\lambda$ is an eigenvalue of the linear operator $S^{-1}TS$

b) Prove that $S^{-1}V_\lambda$ is the $\lambda$-eigenspace of $S^{-1}TS$

c) Prove that $S(V_\lambda)\subseteq V_\lambda$

I don't understand how to find eigenvalues and eigenspaces with respect to linear operators. What changes do I need to make to the methods used with matrices, or do I need a different method entirely. Please help.

user2162882
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Not much changes for doing eigenvalues with general linear operators, for this sort of question. I'll work out part (a) in detail, and leave the others for you to try. For example, the definition is the same:

$\lambda$ is an eigenvalue of $T \iff$ there exists a $v \ne 0$ such that $Tv = \lambda v$.

So you want to find a vector $w$ for which $$S^{-1} T S w = \lambda w$$

Since all we know is that $Tv = \lambda v$, we should use this: It would be nice if we had $Sw = v$; but we can do exactly this by selecting $w = S^{-1} v$. Then we see that

\begin{align*} S^{-1} T S w &= S^{-1} Tv \\ &= S^{-1} \lambda v \\ &= \lambda S^{-1} v \\ &= \lambda w \end{align*} exactly as desired. As one last thing to check, we want to make sure that $w \ne 0$; but this is immediate from the fact that $v \ne 0$.