I recently learned about Kac-Murdock-Szegö matrices, which are given by the pattern
$$ A= \pmatrix{ 1 &r & \cdots &\cdots &r^m\\ r & 1 &r &\cdots &r^{m-1}\\ \vdots &\ddots &\ddots &\ddots &\vdots\\ r^{m} &r^{m-1} & \cdots &r &1 } $$ with very nice properties of its inverse etc.
Unfortunately, the matrix I am analyzing is a bit different. Basically, I get the same structure as A, but with a different exponent:
$$ B= \pmatrix{ 1 &r & r^4 &\cdots &r^{(m^2)}\\ r & 1 &r &\cdots &r^{(m-1)^2}\\ \vdots &\ddots &\ddots &\ddots &\vdots\\ r^{m^2} &r^{(m-1)^2} & \cdots &r &1 } $$ Basically, every exponent is squared. I am interested in properties regarding the determinant and the inverse of such matrices.
When searching for generalizations of Kac-Murdock-Szegö matrices, I can find a few references like e.g. this one, but they all seem to talk about different cases not related to mine.
Does the matrix B have a name I am currently not aware of? Or do you know of any other sources where this kind of matrix is discussed?
Many thanks in advance!
UPDATE FOR SOME CONTEXT AS REQUESTED IN THE COMMENTS:
The matrix appears if you calculate the covariance of an RBF kernel on an equi-spaced grid in the context of Gaussian process regression. These matrices are known for their very bad conditioning numbers (I believe both the smallest and largest eigenvalue grow/decay exponentially), thus it would be practically very relevant to have an analytic solution to calculate the inverses. Kac-Murdock-Szegö matrices do this for the exponential kernel, but unfortunately, the RBF kernel (aka squared exponential) has an additional square, that seems to change the problem significantly, at least in my eyes. Given the beautiful structure, I was hoping that some mathematician already came across it in some other application.