Matrix notation where each element is a quadratic form

Question

I have a matrix of the form,

$$ \begin{bmatrix} x_1^\top A x_1, \ldots, x_1^\top A x_n \\ \cdots, \cdots, \cdots, \\ x_m^\top A x_1, \ldots, x_m^\top A x_n \end{bmatrix} $$

where $A$ is some matrix, and $x_i$'s are column vectors with the appropriate dimensions for the matrix $A$. Note that each element of the above matrix is a scalar in the quadratic form $x_i^\top A x_j$.

Question: Is there a nice compact notation to express the above?

I thought about using Kronecker product; say $B \otimes C $ has elements that look like $b_{ij} C$, which is close to what I'm looking for, except that I have a scalar $b_{ij}$ multiplying the matrix $C$ instead of a vector-times-matrix product.

Thanks!

Answer 1

You can describe this as

$$ \begin{pmatrix} x_1^T \\ \vdots \\ x_m^T \end{pmatrix} A \begin{pmatrix} x_1 & \dots & x_n \end{pmatrix} = \begin{pmatrix} x_1^T \\ \vdots \\ x_m^T \end{pmatrix} \begin{pmatrix} Ax_1 & \dots & Ax_n \end{pmatrix}. $$

Let's say $A \in M_{l \times l}(\mathbb{F})$. The matrix $\begin{pmatrix} x_1 & \dots & x_n \end{pmatrix}$ is the $l \times n$ matrix whose $i$-th column is $x_i$.

$$ \begin{pmatrix} x_1^T \\ \vdots \\ x_m^T \end{pmatrix} $$

is a $m \times l$ matrix whose $i$-th row is $x_i^T$. By block decomposition, the product is your required matrix.