I am interested in studying the determinant and the spectral decomposition of a matrix which is close to being singular. More in particular, the matrix elements are of the form $$ a_{tr} = \frac{1}{t+r}, \quad t,r=1,\dots,T $$ and I would like to complement my numerical analysis, whose details I am reporting below, with some analytical results, but so far I have been unable to do so. To be more precise, I would like to (ideally) have an analytical results for the eigenvalues and the determinant of this matrix.
By numerical analysis I already found that its eigenvalues are exponentially decreasing, with an exponent that for the small eigenvalues scales linearly but actually (and it becomes evident at large eigenvalues only) it scales quadratically with $t$. Also its condition number explodes exponentially as we increase $T$.
Thank you for your kind help!