Let $T$ be a self-adjoint operator on a finite-dimensional inner product space $V$ with distinct eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_k$. Let $E_i$ be the eigenspace of $T$ corresponding to $\lambda_i$, and let $T_i$ be the orthogonal projection of $V$ on $E_i$, $1 \le i \le k$.