Consider the matrix $A$ whose elements are $A_{ij} = a^{|i-j|}$ for $-1 and $i,j=1,\dots,n$
e.g. for $n=4$ the matrix is
$A = \left[ \begin{matrix} 1 & a & a^2 & a^3 \\ a & 1 & a & a^2 \\ a^2 & a & 1 & a \\ a^3 & a^2 & a & 1 \end{matrix} \right]$
Is this matrix always positive definite? If so, what is the simplest way to see that?
I strongly suspect that the matrix is positive definite for all $n$ and $a$, but am having trouble coming up with a proof.
Extra credit: The eigenvalues seem to lie within an interval $[\lambda_{\rm min}, \lambda_{\rm max}]$ which is a function of $a$ but not of $n$. For example, for $a=1/2$ all eigenvalues lie in $[1/3, 3]$.
Is it true for for general $a$, the eigenvalues lie in $[\lambda(a)^{-1}, \lambda(a)]$? If so, what is $\lambda(a)$?
Also, the eigenvectors seem to have a particularly regular form. In particular, they look like that could be expressed as simple combinations of trigonometric functions. Is this the case?