Is there a general formula for $x^{T}Ax \cdot y^{T}Ay$?

Question

Let $x,y \in \mathbb{R}^{n}$. Let A be a $n \times n$ positive-definite symmetric matrix. Is there a general formula for $x^{T}Ax \cdot y^{T}Ay$?

For example, let $x = \begin{bmatrix} 2 \\ 2 \end{bmatrix}, \hspace{0.2cm} y = \begin{bmatrix} 3 \\ 3 \end{bmatrix}, \hspace{0.2cm} A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

Then $x^{T}Ax \cdot y^{T}Ay = 8 \cdot 18 = 144$

This is equal to $2(x \circ y)^{T}A(x \circ y)$ where $(x \circ y) = \begin{bmatrix} 6 \\ 6 \end{bmatrix}$ is the Hadamard product of $x,y$.

It seems that $x^{T}Ax \cdot y^{T}Ay = 2(x \circ y)^{T}A(x \circ y)$ also works for $x = \begin{bmatrix} 5 \\ 5 \end{bmatrix}, \hspace{0.2cm} y = \begin{bmatrix} 7 \\ 7 \end{bmatrix}$. Is this true in the general case, and if so, how do I prove it? If not, how can I find and prove a general formula?

Answer 1

The Hadamard product won't help, but the Kronecker product distributes over the matrix product, so one can write $$\eqalign{ \left(x^TAx\right)\left(y^TAy\right) &= \left(x^TAx\right)\otimes\left(y^TAy\right) \\ &= (x\otimes y)^T\left(A\otimes A\right)(x\otimes y) \\ }$$ To start things off, the product between the two scalar expressions can also be replaced by a Kronecker product.