[From Wikipedia:][1]

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

Then the Min-Max principle states:

$$\lambda _{1}\leq R_{A}(x)\leq \lambda _{n}\quad \forall x\in {\mathbf {C}}^{n}\backslash \{0\} \quad (*)$$

where $\lambda_1, \lambda_n$ are the least and largest eigenvalues of $A$ respectively.

In simpler form: $\lambda_1\|x\|^2 \leq x^TAx \leq \lambda_n\|x\|^2$

Consider instead $x,y \in {\mathbf{C}}^{n}\backslash \{0\}$, 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 $(*)$ hold for $R_{A}(x,y)$? [1]: https://en.wikipedia.org/wiki/Min-max_theorem

From Wikipedia:

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

[From Wikipedia:][1]

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

Then the Min-Max principle states:

$$\lambda _{1}\leq R_{A}(x)\leq \lambda _{n}\quad \forall x\in {\mathbf {C}}^{n}\backslash \{0\} \quad (*)$$

where $\lambda_1, \lambda_n$ are the least and largest eigenvalues of $A$ respectively.

In simpler form: $\lambda_1\|x\|^2 \leq x^TAx \leq \lambda_n\|x\|^2$

Consider instead $x,y \in {\mathbf{C}}^{n}\backslash \{0\}$, 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 $(*)$ hold for $R_{A}(x,y)$? [1]: https://en.wikipedia.org/wiki/Min-max_theorem

[From Wikipedia:][1]

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

Then the Min-Max principle states:

$$\lambda _{1}\leq R_{A}(x)\leq \lambda _{n}\quad \forall x\in {\mathbf {C}}^{n}\backslash \{0\} \quad (*)$$

where $\lambda_1, \lambda_n$ are the least and largest eigenvalues of $A$ respectively.

In simpler form: $\lambda_1\|x\|^2 \leq x^TAx \leq \lambda_n\|x\|^2$

Consider instead $x,y \in {\mathbf{C}}^{n}\backslash \{0\}$, 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 $(*)$ hold for $R_{A}(x,y)$? [1]: https://en.wikipedia.org/wiki/Min-max_theorem