I am working on a problem were I encounter matrices of the form

$A = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$

I am aware of Cauchy matrices, which have the form

$A = \begin{bmatrix}\frac{1}{a_i - b_j}\end{bmatrix}_{ij}$

(sometimes written with a plus rather than a minus). Many of the results I need I can actually obtain by factoring the above matrix as a product of a diagonal matrix with a Cauchy matrix (assuming the $a_i \neq 0$), as in:

$A = \mathbb{diag}(a_i^{-1})\begin{bmatrix}\frac{1}{a_i^{-1} - b_j}\end{bmatrix}.$

These matrices arise as the solutions to matrix equations of the form

$X - AXB^T = C$

which are discrete-time analogs of Sylvester equations:

$AX + XB = C.$

(Also, related are Lyapunov equations and algebraic ricatti equations). It seems that these must appear in the literature somewhere, but I haven't been able to find them. My question is:

>1. Do matrices of the form $A = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$ have a name in the literature?

> 2. Is anyone aware of good references for general results on these matrices? For example, there are general results on the determinant and inverses of Cauchy matrices.

As I mentioned, I have already found a determinant formula and a formula for the inverse of the matrix using the factorization I mentioned above. But it would be helpful to know of further results if they exist and I would like to properly cite the literature as well.