While exploring some data modelling, I observed without being able to prove it that the inverse of matrix $A$ of the following form:
- it is hermitian, i.e. $A_{ij}=A^{*}_{ji}$
- it has non-zero positive elements over the diagonal, i.e. $A_{ii}>0$
- it has non-zero constant sub-diagonals $A_{kl} = f(k-l)$ with $f$ an arbitrary function of integer variable $n$, non-zero only for a few $n$
- it is zero elsewhere
is itself invertible and its inverse is within the same class of matrices. My questions:
Are these matrices part of a well-known class?
Is there any chance an inversion algorithm (or an analytical expression for the inverse) is available in the literature?
Please note that this is close but not equal to a sparse Toeplitz matrix, since the diagonal entries are all different, a priori.