Direct sum of intersections of Eigenspaces

Question

Let $f,g$ be diagonalizable Endomorphisms of a finite dimensional $\mathbb{K}$-Vectorspace with $f\circ g=g\circ f$. Let $\lambda_{1},...,\lambda_{k}$ and $\mu_{1},...,\mu_{l}$ be the eigenvalues of $f$ and $g$ respectively.

Show that:

$Eig(g,\mu_{j})=(Eig(f,\lambda_{1})\cap Eig(g,\mu_{j}))\oplus....\oplus(Eig(f,\lambda_{k})\cap Eig(g,\mu_{j}))$

I already know that $g(Eig(f,\lambda_{i}))\subseteq Eig(f,\lambda_{i})$

Since this is a homework problem I would like to recieve only hints and no full solution.

Also: Does this theorem have any bigger implication or usage in Linear Algebra ?

Answer 1

Hint: The restriction of $f$ to $g(Ei(f,\mu_i)$ is diagonalizable and its eigenspaces are $Ei(f,\lambda_k)\cap Ei(g,\mu_i)$.