I've asked this question at MathOverflow and was told it'd be better suited for here.

In Paul A. Clement's (1959) paper,

> 
> [A Class of Triple-Diagonal Matrices for Test Purposes](http://dx.doi.org/10.1137/1001006), *SIAM Review*, Vol. 1, No. 1 (Jan., 1959), pp. 50-52
> 
> 
> 

He makes the claim that the eigenvalues of:

$ \begin{pmatrix}
0 & y\_{1} & 0 & ... & 0 \\\ 
x\_{1} & 0 & y\_{2} & & ... \\\
0 & x\_{2} & 0 & ... & 0 \\\ 
... & & ... & & y\_{n} \\\ 
0 & ... & 0 & x\_{n} & 0
\end{pmatrix} $

are $\pm (n), \pm (n-2), ..., (\pm1 \; or \; 0)$ for $x\_{k} = k$ and $y\_{k} = n-k+1$.

Specifically, and I quote, "then a theorem of Sylvester establishes that the eigenvalues of this An+, are the
numbers".

I can't for the life of me figure out what theorem and/or how it follows from them. I am familiar with Sylvester's formula for matrices in terms of their eigenvalues, but to get Frobenius covariants of a matrix A one needs to know the eigenvalues to start with.

Am I overlooking something trivial here?