The matrix whose positive definiteness I am investigating is the following one $$ H_k(p)=\begin{bmatrix}1 & \frac{1}{p+1} & \frac{1}{p+2} & \frac{1}{p+3} & ... & \frac{1}{p+k-1} \\ \frac{1}{p+1} & \frac{1}{p+2} & \frac{1}{p+3} & ... & ...& \frac{1}{p+k}\\ \vdots & \ldots &\ldots &\ldots &\ldots &\vdots \\ \vdots & \ldots &\ldots &\ldots &\ldots &\vdots \\ \frac{1}{p+k-1} & \ldots &\ldots &\ldots &\ldots &\frac{1}{p+2(k-1)} \end{bmatrix} $$ where $p$ is any positive integer and $k$ is the size of the Hankel Matrix
It is clear for me that this matrix is not positive definite if $p$ is not a positive integer: as a counterexamples it sufficient to consider its determinant when $k=2$. And indeed calculating that determinant I got the interval of values $p$ for which it is negative.
Nevertheless I would like to know if this matrix becomes positive definite if I take into account that $p$ is only a positive integer, and to settle the problem I tried reducing the matrix to the row echelon form to get the eigenvalues, but I didn't get far.
I would really like to get some hints in order to proceed.
Yes, each row (or column) is not a linear combination of the others, thus its invertible. It is obviously symmetric. You can put it in row echelon form (with Gaussian eliminiation) and see that the diagonal remains positive, thus all eigenvalues are positive, so it is then positive definite.