Let $\displaystyle B$ be a $\displaystyle nxn$ matrix and $\displaystyle V_{\lambda} =\{v \in \mathbb{C}^n : Bv=\lambda v\}$. Let $\displaystyle \{e_1^\lambda ,e_2^\lambda, ... ,e_{k(\lambda)}^\lambda\}$ be an orthonormal basis for $\displaystyle V_\lambda$ consisting of eigenvectors for $\displaystyle C_\lambda$ , where$\displaystyle C_\lambda$ is a self-adjoint linear transformation on $\displaystyle V_\lambda$.

Prove that $\displaystyle \{e_i^\lambda : \lambda$ is an eigenvalue for $\displaystyle B$ and $\displaystyle 1\le i\le k(\lambda)\}$ is an orthonormal basis for $\displaystyle \mathbb{C}^n$.