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?



Inverse of an AR(1) or Laplacian (?) or Kac-Murdock-Szegö matrix
Your $C\_{(i,j)(k,l)}$ is the Kronecker product of two $C$ matrices (with different constant $\alpha\_x$ and $\alpha\_y$. And $(A \otimes B)^{-1}=A^{-1}\otimes B^{-1}$.


