**[From Wikipedia:][1]**

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 $R_{A}(x,y)$?

  [1]: https://en.wikipedia.org/wiki/Min-max_theorem