I have a little question regarding diagonalizable endomorphisms:
Let $V$ be an finite-dimensional vectorspace of some field $K$. Let $f,g:V\rightarrow V$ be diagonalizable endomorphisms such that $f\circ g=g \circ f$. We denote the eigenvalues of $f$ and $g$ with $\lambda_1,\dots,\lambda_k$ and $\mu_1,\dots,\mu_l$:
If $A_f^B$ and $A_g^B$ are the associated matrices of $f$ and $g$ with repsect to the basis $B$, show that there exists a basis $B$ of $V$ such that $A_f^B$ and $A_g^B$ are in a diagonal form. Unfortunately I don't quite know where to start. I know that it's equivalent to there being a basis $B$ of $V$ such that I'm able to choose $dim(V)$ linear independant vectors such that each vector is element of both eigenspaces $E(A_f^B,\lambda_i)$ and $E(A_g^B,\mu_j)$ for some $i$ and $j$.
Does anybody have an idea?
Thanks for checking in ;)
~Cedric
