Question
$S$ is a $2^{n} \times n$ matrix where the rows are binary representations of 0 to $2^{n} - 1$.
For $n = 2$,
$$S = \begin{bmatrix} 0 & 0 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \\ \end{bmatrix}$$
$\vec{x} = \begin{bmatrix}x_{1} & x_{2}& \ldots & x_{n}\end{bmatrix}$ is a real vector with elements in interval $(0, 1)$
$A$ is a $2^{n} \times n$ matrix where
$$A_{ij} = \begin{cases} 1/x_j & \text{if } S_{ij} = 0 \\ 1/(1 - x_j) & \text{if } S_{ij} = 1 \\ \end{cases}$$
What is $s_{n} = \det(A^{\intercal}A)$?
Thank you.
If $C=A^tA$ then
$$C_{ii}=2^{n-1}\left(\frac1{x_i^2}+\frac1{(1-x_i)^2}\right)\\ C_{ij}=2^{n-2}\left(\frac1{x_i}+\frac1{1-x_i}\right)\left(\frac1{x_j}+\frac1{1-x_j}\right)$$ So $C$ is the sum of a diagonal matrix and a rank-1 matrix.