Let $H$ be a finite-dimensional complex [Hilbert space](/page/Hilbert%20Space) and let $A:H\to H$ be self-adjoint. Then there are distinct [real numbers](/page/Real%20Numbers) $a_1,\dots,a_m$ and mutually orthogonal projections $P_1,\dots,P_m$ such that \begin{align*} A = \sum_{k=1}^m a_k P_k, \qquad I = \sum_{k=1}^m P_k. \end{align*}