# Self-Adjoint Operators Basis of Eigenfunctions [duplicate]

I encountered the following statement in the context of the Hecke algebra on the space of cusp forms, left without proof:

Let there be a family of commutative self-adjoint operators on a finite dimensional vector space $$V$$. Then there exists a basis of $$V$$ consisting of functions which are eigenfunctions for all the operators.

I would appreciate a proof of this, as elementary as possible, or a reference. I only know basic functional analysis and no spectral theory so I don't know where to start.

Proof: Let $$A,B$$ be the two matrices; let $$T:=A+iB$$; this is a normal matrix ($$T^*T=TT^*$$) hence diagonalisable to $$P^{-1}TP=C+iD$$, hence $$P^{-1}AP=C$$, $$P^{-1}BP=D$$ (note that the real and imaginary parts are unique). Writing $$AP=PC$$ in terms of the column vectors, we find $$Ap_i=c_ip_i$$, and similarly, $$Bp_i=d_ip_i$$, so $$p_i$$ are a common set of eigenvectors for $$A,B$$.
Proof: They all commute with one of them $$A$$, hence have the same eigenvectors.