Spectrum of the matrix $A=(a_{ij})$ where $a_{ij}=i+j$ What is the spectrum of the matrix $A=(a_{ij})_{n\times n}$ where $a_{ij}=i+j$ for any $n$. Also, what are the eigenvectors corresponding to their eigenvalues? 

Progress. This matrix is definitely symmetric. Hence it contains only real eigenvalues. Also it will contain only one positive and one negative eigenvalue and the rest are zero eigenvalues. The sum of the eigenvalues is $n(n+1)$. 

Hints:
Note that the matrix $A$ has rank $2$.  So, for $n>2$, $A$ will have eigenvalue $0$ with multiplicity $n-2$.
Note that $A$ is symmetric.  It follows that the kernel of $A$ is the orthogonal complement of its image.  That is, the eigenspace associated with $\lambda = 0$ is the orthogonal complement of the space spanned by the vectors
$$
(1,\dots,1),(1,2,\dots,n)
$$
Moreover, we can necessarily find an orthonormal basis of $\Bbb R^n$ consisting of eigenvectors, and all eigenvalues will be real.
We can find the remaining $2$ eigenvalues $\lambda_1,\lambda_2$ by noting that
$$
\operatorname{trace}(A) = \lambda_1 + \lambda_2\\
\operatorname{trace}(A^2) = \lambda_1^2 + \lambda_2^2
$$
Alternatively, it suffices to find the eigenvalues of $X^*AX$, where the columns of $X$ are any orthonormal basis for the span of $\{(1,\dots,1),(1,2,\dots,n)\}$.
