Courant-Fischer min-max principle for self-adjoint endomorphisms on finite-dimensional real inner product spaces

By [L1], choose an orthonormal eigenbasis $(e_1,\dots ,e_n)$ with $T{e}_j={\lambda }_j{e}_j$. For every nonzero $v={\sum }_j{c}_j{e}_j$, one has $R_T(v)=\frac{{\sum }_j{\lambda }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2}{{\sum }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2}$, so on $U_k:=\mathrm{span}(e_1,\dots ,e_k)$ every Rayleigh quotient is at least ${\lambda }_k$ and on $W_k:=\mathrm{span}(e_k,\dots ,e_n)$ every Rayleigh quotient is at most ${\lambda }_k$; equality holds at $e_k$ in both cases.

Let $W\subseteq V$ have dimension $n-k+1$. If $W\cap U_k=\{0\}$, then $W\subseteq U_k^{\perp }=\mathrm{span}(e_{k+1},\dots ,e_n)$, whose dimension is $n-k$, contradicting [L2]. Thus some nonzero $v\in W\cap U_k$ satisfies $R_T(v)\ge {\lambda }_k$ by step 1.1, so ${\max }_{0\mathrm{\ne }v\in W}{R}_T(v)\ge {\lambda }_k$. Since step 1.1 also gives ${\max }_{0\mathrm{\ne }v\in W_k}{R}_T(v)={\lambda }_k$, the first Courant-Fischer equality follows.

Let $U\subseteq V$ have dimension $k$. If $U\cap W_k=\{0\}$, then $U\subseteq W_k^{\perp }=\mathrm{span}(e_1,\dots ,e_{k-1})$, whose dimension is $k-1$, contradicting [L2]. Thus some nonzero $v\in U\cap W_k$ satisfies $R_T(v)\le {\lambda }_k$ by step 1.1, so ${\min }_{0\mathrm{\ne }v\in U}{R}_T(v)\le {\lambda }_k$. Since step 1.1 also gives ${\min }_{0\mathrm{\ne }v\in U_k}{R}_T(v)={\lambda }_k$, the second Courant-Fischer equality follows.