Determinant and eigenvalues of a specific matrix This came up in a conversation with an engineer friend of mine.
Let $c>0$ be a constant. Let $A_{ij}$ be an $n$ by $n$ matrix with entries
$$
A_{ij} = e^{-c(i-j)^2}.
$$
Is there a name for this matrix? What is known, perhaps approximately or asymptotically as 
$c$ and $n$ change about the determinant and the eigenvalues of it? Are there other functions
of $(i-j)$ for which the answer is more explicit?
 A: To answer the question: "...is there a name for this matrix..."
Beyond the "Gaussian-Toeplitz matrix" mentioned in the comments, the said matrix is a special case of the Gaussian Kernel (which is also the alluded to "heat kernel"):
\begin{equation*}
 k(x_i,x_j) = e^{-c(x_i-x_j)^2}\qquad x_1,\ldots,x_n \in \mathbb{R}.
\end{equation*}
Clearly, the matrix $A = [k(x_i,x_j)]$ is semidefinite. If the $x_i$s are unique, then it is strictly positive definite (this requires some more work to prove).
The Gaussian kernel is of course the canonical example of a translation invariant kernel, so searching more for such kernel matrices should bring up more examples, namely, matrices of the form $A_{ij} = k(x_i,x_j)=\varphi(x_i-x_j)$, for suitable $\varphi$.
Incidentally, in the paper mentioned by @Lucia, the smallest eigenvalue for the Gaussian-Toeplitz $A$ is bounded from below by (assuming $A$ is $n\times n$):
\begin{equation*}
  \lambda_n \ge \frac{\det(A)}{n!}.
\end{equation*}
It seems possible to improve this to $\lambda_n \ge \frac{\det(A)}{2^{n-1}}$ by using the technique mentioned in my answer here.
A: Just a followup to @Lucia's comment, here is the paper:
