Eigenvalues of a symmetric matrix with each entry being the product of the indices

Question

We have a matrix $A$ such that $A_{ij}=ij$, and we've been asked to find its largest eigenvalue if it's an $n\times n$ matrix. I computed some of these and it turned out to be the square pyramidal numbers ($5,14,30,55,\ldots$) but how does one go about proving this?

Answer 1

First, note that $A=\begin{bmatrix}1\\\cdots\\n\end{bmatrix}\begin{bmatrix}1&\cdots&n\end{bmatrix}$.

Therefore, $A$ has rank 1 (i.e. 0 has multiplicity $n-1$) and has only one nonzero eigenvalue.

This is exactly the trace, which has value $\sum_{k=1}^n k^2$.