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I am trying to do the following exercise:

Let $V$ be a $K$-finite dimensional vector space and let $f,g \in Hom(V,V)$. Define $Spec(f)=\{\alpha \in K / \alpha \space \text{is an eigenvalue of f}\}$. Prove that $Spec(f \circ g)=Spec(g \circ f)$.

Suppose one of the two linear transformations is an isomorphism, w.l.g., suppose $g$ is an isomorphism, then if $|f|_{\mathcal B}=A, |g|_{\mathcal B}=B$ for some basis $\mathcal B$ of $V$, then the characteristic polynomial of $AB$ is $$\mathcal X_{AB}=det(\alpha I_n - AB)$$$$=det(B(\alpha I_n - AB)B^{-1})$$$$=det(\alpha I_n - BA)=\mathcal X_{BA}$$

From here it follows $\alpha$ is an eigenvalue of $f \circ g$ if and only if it is an eigenvalue of $g \circ f$.

I don't know what to do for the general case. Any help would be appreciated.

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A more direct proof would be:

Assume $\lambda$ is a non-zero eigenvalue of $f\circ g$, i.e.

$$ (f\circ g)v=\lambda v$$ for some $0\neq v\in V$.

Application of $g$ on both sides shows

$ (g\circ f) (g(v)) =\lambda g(v)$

which shows that $\lambda$ is also an eigenvalue of $g\circ f$ (with eigenvetor $g(v)$, which is not equal to $0$, otherwise $(f\circ g)v=\lambda v$ would be zero, a contradiction to $\lambda,v\neq 0$).

If $\lambda=0$ is an eigenvalue of $f\circ g$, then $\text{det}(f\circ g)=0$, but then also $\text{det}(g\circ f)=0$ so $g\circ f$ is also not-injective, i.e. has eigenvalue $0$.

This shows that every eigenvalue of $f\circ g$ is also one of $g\circ f$. The other inclusion works the same way.

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This works not just for square matrices but all matrices where the product makes sense. A reference is Godsil and Royle's book Algebraic graph theory, Lemma 8.2.4. Their ingenious proof is to note that $$ \det\left(\begin{bmatrix}I & A \\ B & I\end{bmatrix} \begin{bmatrix}I & 0 \\ -B & I\end{bmatrix}\right) = \det\left(\begin{bmatrix}I & 0 \\ -B & I\end{bmatrix} \begin{bmatrix}I & A \\ B & I\end{bmatrix}\right). $$ The multiplicities of the eigenvalues will, in general, be different.

The result is used in graph theory to relate the number of spanning trees of a graph to that of its line graph.

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