My current problem involves having an exact (symbolic) inverse of a scaled AR(1) matrix for n-dimension. (I don't know what this matrix would be called in general; I'm sure it is used often.) This is used as a smoothing prior on a function sampled on a uniform grid. For a 1-dimensional function, the matrix is
$C = \rho \begin{bmatrix} 1 & \alpha & \alpha^2 & \cdots & \alpha^n \\ \alpha & 1 & \alpha & \cdots & \alpha^{n-1}\\ \alpha^2 & \alpha & 1 & \\ \vdots & \vdots & & \ddots & \\ \alpha^n & \alpha^{n-1} & & & 1 \end{bmatrix} $
and I know the inverse which is,
$ C^{-1} = \frac{1}{\rho(1-\alpha^2)} \begin{bmatrix} 1 & -\alpha & & & 0\\ -\alpha & 1+\alpha^2 & - \alpha \\ & -\alpha & 1+\alpha^2 &\ddots \\ & &\ddots & \ddots & -\alpha \\ 0 & & & -\alpha & 1 \end{bmatrix} $
(This can be also found in Kac, M., Murdock, W., and Szegö, G. (1953). On the eigenvalues of certain Hermitian forms. J. Rational Mech. Anal, 2:767–800.)
I would imagine that this can be generalized to higher dimensional case where the $\alpha$ now spreads in each direction. This would allow my uniformly sampled n-dimensional function to be smooth. Sort of having the form \begin{equation} C_{(i,j),(k,l)} = \rho \alpha_x^{|i-k|} \alpha_y^{|j-l|}, \end{equation} But as a giant matrix for flattened function (the vec-operation; representing the n-dimensional function as a vector with some ordering). Can anybody recommend a book on such symbolic matrix inversions that would have this?
