Return to Question

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

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

I am aware of Cauchy matrices, which have the form

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

(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:

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

These matrices arise when computing 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 Riccati 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 $X = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$ 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.

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

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

I am aware of Cauchy matrices, which have the form

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

(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:

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

These matrices arise when computing 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 Riccati 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 $X = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$ 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.

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

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

I am aware of Cauchy matrices, which have the form

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

(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:

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

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 Riccati 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.

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

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

I am aware of Cauchy matrices, which have the form

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

(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:

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

These matrices arise when computing 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 Riccati 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 $X = \begin{bmatrix}\frac{1}{1 - a_ib_j}\end{bmatrix}_{ij}$ 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.

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.

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

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

I am aware of Cauchy matrices, which have the form

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

(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:

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

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 Riccati 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.