Commuting Kraus operators are normal

Let $\{V_1,\dots ,V_n\}$ be a set of mutually commuting matrices. We show that if $V_1^{\ast }{V}_1+\cdots +V_n^{\ast }{V}_n=\mathrm{I}\mathrm{d}$ then the matrices are normal and, in particular, simultaneously diagonalizable.