Courant–Fischer min-max principle

For a compact positive self-adjoint operator $T$ with decreasing eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \cdots$, the Courant–Fischer principle givesIn particular, $S\ge T$ in quadratic-form order implies ${\lambda }_k(S)\ge {\lambda }_k(T)$ for every $k$.