The inverse of AR structure correlation matrix / Kac-Murdock-Szegő matrix I want to find the inverse of the following matrix:
$$ R_{k-1} = 
   \begin{pmatrix}
    1 &\rho &\rho^2 & \dots &\rho^{k-2} \\
    \rho &1 &\rho   & \dots  &\rho^{k-3} \\
    \rho^2 &\rho &1 & \dots&\rho^{k-4} \\
    \vdots &\vdots &\vdots &\ddots &\vdots\\ 
    \rho^{k-2} & \rho^{k-3} & \rho^{k-4} & \dots & 1\end{pmatrix}$$
Let $A_{i,j}$ be the $i,j$ minor of $R_{k-1}$. By considering the pattern of the above matrix and its symmetrical properties, we can conclude that:

*

*$\det(A_{11})  = \det(A_{k-1,k-1})= |R_{k-2}|$

*$\det(A_{i,j}) = \det((A_{j,i})^T)$

*$\det(A_{i,j}) = 0$ for $|i-j|\le2$
which means that the inverse of $R_{k-1}$ is a tridiagonal symmetric matrix. I've tried to find the inverse using the fact I've described above and using  $A^{-1}A=A A^{-1}=I$. But I couldn't find it, since there are more variables than equations. Did i miss something?
or may be is there any other easier way to find the inverse?
 A: When $\rho$ is close to zero, $R_{k-1}$ is positive definite. Therefore we may perform a Cholesky decomposition and obtain $R_{k-1}=L\,\operatorname{diag}(1,\,1-\rho^2,\ldots,\,1-\rho^2)\,L^T$, where
$$
L=
\pmatrix{
1\\
\rho       &1\\
\rho^2     &\rho       &1\\
\vdots     &\ddots     &\ddots &\ddots\\
\vdots     &\ddots     &\ddots &\ddots &1\\
\rho^{k-2} &\rho^{k-3} &\cdots &\rho^2 &\rho &1}.
$$
It follows that
\begin{align}
R_{k-1}^{-1}
&=(L^{-1})^T\,\operatorname{diag}\left(1,\frac1{1-\rho^2},\ldots,\frac1{1-\rho^2}\right)\,L^{-1}\\
&=\frac1{1-\rho^2}(L^{-1})^T\,(-\rho^2E_{11}+I)\,L^{-1}
\end{align}
with
$$
L^{-1}=
\pmatrix{
1\\
-\rho&1\\
&\ddots &\ddots\\
&&\ddots &\ddots\\
&&&-\rho &1}.
$$
Consequently,
$$
\renewcommand{\d}{\phantom{1}-\rho\phantom{^2}}
R_{k-1}^{-1}=\frac1{1-\rho^2}
\pmatrix{
1      &\d\\
\d &1+\rho^2 &\d\\
&\d &\ddots&\ddots\\
&&\ddots &\ddots&\ddots\\
&&&\ddots &1+\rho^2 &\d\\
&&&&\d&1}.
$$
A: Sometimes the easiest way to deduce something is through an experiment, which works here quite well.
As a candidate, try the tridiagonal matrix with the entries
$$
\frac{1}{1-\rho^2},\frac{1+\rho^2}{1-\rho^2},\ldots,\frac{1+\rho^2}{1-\rho^2},\frac{1}{1-\rho^2}
$$
on the diagonal and the entries
$$
-\frac{\rho}{1-\rho^2}
$$
above and below the diagonal. Of course, $\rho$ must be different from one.
