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

```
A Class of Triple-Diagonal Matrices for Test Purposes
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-1} \\\ 0 & ... & 0 & x_{n-1} & 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?