**[From Wikipedia:](https://en.wikipedia.org/wiki/Min-max_theorem)**


Let $A$ be an $n \times n$ Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient $R\_A : 
\mathbf C^n \setminus \{0\} \to \mathbb{R}$ defined by
$$R\_{A}(x)={\frac{(Ax,x)}{(x,x)}}$$ where $(\cdot, \cdot)$ denotes the Euclidean inner product on $\mathbf C^n$.


**Then the Min-Max principle states:**



> 
> $$\lambda \_{1}\leq R\_{A}(x)\leq \lambda \_{n}\quad \forall x\in
>  {\mathbf {C}}^{n}\setminus \{0\}.\tag{$\*$}\label{star}$$
> 
> 
> 


where $\lambda\_1, \lambda\_n$ are the least and largest eigenvalues of $A$ respectively.


In simpler form: $\lambda\_1\lVert x\rVert^2 \leq x^TAx \leq \lambda\_n\lVert x\rVert^2$.




---


Consider instead $x,y \in {\mathbf{C}}^{n}\setminus \{0\}$, with $x \neq y$ and the Rayleigh quotient defined as:
$$R\_{A}(x,y)={\frac{(Ax,y)}{(x,y)}}.$$


On what condition on the vectors $x$, $y$ does \eqref{star} hold for $R\_{A}(x,y)$?



Under what condition does Courant–Fischer–Weyl min-max principle hold in general?
To avoid the various ambiguities in your statement I'll assume that $A$ is *real* and symmetric. $\newcommand{\bR}{\mathbb{R}}$ Choose an *orthonormal* basis $\newcommand{\be}{\mathbf{e}}$ $\be\_1,\dotsc,\be\_n$ consisting of eigenvectors of $A$ and denote by $x\_1,\dotsc, x\_n$ the coordinates determined by this basis. if $x,y\in\bR^n$ are vectors such that $x\_iy\_i\geq 0$ and $(x,y)>0$, then the above equalities hold. Indeed, if $\lambda\_1\leq \cdots \leq \lambda\_n$ are the eigenvalues of $A$, then


$$ (Ax,y)=\sum\_{i=1}^n \lambda\_i x\_iy\_i\geq \lambda\_1\sum\_{i=1}^n x\_iy\_i=\lambda\_1(x,y). $$


Note that the conditions $x\_iy\_i\geq 0$ can be wriiten invariantly as


$$(x,\be)(y,\be)\geq 0 $$


for any eigenvector $\be$ of $A$.


