I could be very confused (likely), but notice that your matrix the sum of $N$ and $N^t,$ where $N$ is the upper triangular matrix where $N_{ij}= 1/i,$ when $i<j$ and $0$ otherwise.

It seems that by Denis Serre's answer to [this question][1], the answer to yours is YES.


  [1]: http://mathoverflow.net/questions/52578/eigenvalues-of-sum-of-a-non-symmetric-matrix-and-its-transpose-aat