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Let $A$ be an $n\times n$ matrix, with $A_{ij}=i+j$. Find the eigenvalues of $A$. A student that I tutored asked me this question, and beyond working out that there are 2 nonzero eigenvalues $a+\sqrt{b}$ and $a-\sqrt{b}$ and $0$ with multiplicity $n-2$, I’m at a bit of a loss.

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Let $u=(1,1,\ldots,1)^T$ and $v=(1,2,\ldots,n)^T$. Then $A=uv^T+vu^T$. As $u$ is not parallel to $v$ and they are nonzero vectors, $A$ has rank 2 and every eigenvector for a nonzero eigenvalue $\lambda$ must lie in the span of $u$ and $v$. Let $(\lambda,\,pu+qv)$ be such an eigenpair. Then

$$

\lambda(pu+qv)=(uv^T+vu^T)(pu+qv)=u[v^T(pu+qv]+v[u^T(pu+qv)].

$$

Since $u$ and $v$ are linearly independent, by comparing coefficients on both sides, we get

$$

\lambda\pmatrix{p\\ q}=\underbrace{\pmatrix{v^Tu&v^Tv\\ u^Tu&u^Tv}}_B\pmatrix{p\\ q}.

$$

Therefore the nonzero eigenvalues of $A$ are exactly the two eigenvalues of $B$. Since the characteristic polynomial of $B$ is

$$

\lambda^2-2(u^Tv)\lambda+[(u^Tv)^2-(u^Tu)(v^Tv)]

\equiv (\lambda-u^Tv)^2-(u^Tu)(v^Tv),

$$

it follows that

$$

\color{red}{\lambda=u^Tv\pm\sqrt{(u^Tu)(v^Tv)}.}

$$

In your case, by putting $u=(1,1,\ldots,1)^T$ and $v=(1,2,\ldots,n)^T$, we get

$$

\lambda = \frac{n(n+1)}2\pm\sqrt{\frac{n^2(n+1)(2n+1)}6}.

$$

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