Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix $$ X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}. $$ Such matrix arises in the solution of particular Stein matrix equations (e.g., see p. 11 of Bhatia, "Positive definite matrices", Princeton University Press, 2007), and seems to be somehow related to the class of Cauchy matrices. $X$ seems to have a interesting structure (and, I guess, it may feature some "nice" properties), however I couldn't find any additional information on this matrix on the web. So my question:

Does matrix $X$ have a name? Are there any known property of $X$ (such as formulas for the determinant, inverse, etc.)?

P.S. A property that follows from the connection with Stein equations is that $X$ is always positive definite.

Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix $$ X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}. $$ Such matrix arises in the solution of particular Stein matrix equations (e.g., see p. 11 of Bhatia, "Positive definite matrices", Princeton University Press, 2007), and seems to be somehow related to the class of Cauchy matrices. $X$ seems to have a interesting structure (and, I guess, it may feature some "nice" properties), however I couldn't find any additional information on this matrix on the web. So my question:

Does matrix $X$ have a name? Are there any known properties of $X$ (such as formulas for the determinant, inverse, etc.)?

P.S. A property that follows from the connection with Stein equations is that $X$ is always positive semidefinite.