Rayleigh-Ritz Theorem

Definition

Let $A$ be an $n\times n$ Hermitian Matrix with eigenvalues sorted in descending order ${\lambda }_1,\dots ,{\lambda }_p$, where $p\le n$, then $\underset{\mathbf{x}\ne 0}{\max }\frac{{\mathbf{x}}^{⊺}A\mathbf{x}}{{\mathbf{x}}^{⊺}\mathbf{x}}=\underset{||\mathbf{x}|{|}_2=1}{\max }{\mathbf{x}}^{⊺}A\mathbf{x}={\lambda }_1$ $\underset{\mathbf{x}\ne 0}{\min }\frac{{\mathbf{x}}^{⊺}A\mathbf{x}}{{\mathbf{x}}^{⊺}\mathbf{x}}=\underset{||\mathbf{x}|{|}_2=1}{\min }{\mathbf{x}}^{⊺}A\mathbf{x}={\lambda }_p$