The Courant–Fischer Min–Max Theorem

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation and Linear Algebra: Standing Notation, let $n\in\mathbb{N}$, let $A\in\mathcal{S}(n)$ be a symmetric matrix, let $Q\in\mathrm{O}(n)$ and $\lambda\in\mathbb{R}^{n}$ satisfy

$$A=(Q\operatorname{diag}(\lambda))Q^{\top},\qquad\lambda_{1}\ge\lambda_{2}\ge\dots\ge\lambda_{n},$$

as provided by The Spectral Theorem for Real Symmetric Matrices (i), and let $k\in[n]$. A unit vector is an $x\in\mathbb{R}^{n}$ with $\lVert x\rVert=1$.

(i) Max–min. Every subspace $U$ of $\mathbb{R}^{n}$ (ℝⁿ Is a Real Vector Space (i)) with $\dim U=k$ contains a unit vector $x$ with $x\cdot(Ax)\le\lambda_{k}$. The span $U_{k}=\operatorname{span}(Qe_{1},\dots,Qe_{k})$ is a subspace of dimension $k$, and

$$x\cdot(Ax)\ge\lambda_{k}\qquad\text{for every unit vector }x\in U_{k}.$$

(ii) Min–max. Every subspace $U$ of $\mathbb{R}^{n}$ with $\dim U=n-k+1$ contains a unit vector $x$ with $x\cdot(Ax)\ge\lambda_{k}$. The span $V_{k}=\operatorname{span}(Qe_{k},\dots,Qe_{n})$ is a subspace of dimension $n-k+1$, and

$$x\cdot(Ax)\le\lambda_{k}\qquad\text{for every unit vector }x\in V_{k}.$$