Under what condition does Courant–Fischer–Weyl min-max principle hold in general?

Question

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: $\lambda_1\lVert x\rVert^2 \leq x^TAx \leq \lambda_n\lVert x\rVert^2$.

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

Answer 1

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 $A$.