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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, 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?

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    $\begingroup$ There is a typo: your matrix should be of order $n+1$ instead of $n$, i.e. the last index for $x_i$ or $y_i$ is $n$, not $n-1$. For an alternative proof, see my answer in another thread. $\endgroup$ – user1551 May 29 '13 at 12:36
  • $\begingroup$ Fixed it, thanks! $\endgroup$ – FlamingWilderbeest May 29 '13 at 12:44
  • $\begingroup$ In short: Sylvester in fact devoted an entire note to these matrices, and proved a theorem on the eigenvalues. See the papers I linked to in my answer. $\endgroup$ – J. M. is a poor mathematician May 29 '13 at 18:36
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A better discussion of the (Clement-)Kac(-Sylvester) tridiagonal matrices would be in this paper and this paper; there are a few slick (I could not have come up with them, I'll admit) proofs that the eigenvalues of these matrices are those particular integers.

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EDIT: OP has changed the question, so what follows is no longer relevant, and can safely be ignored.

If $n=3$, the matrix is $$\pmatrix{0&3&0\cr1&0&2\cr0&2&0\cr}$$ which has characteristic polynomial $t^3-7t$ and eigenvalues $\pm\sqrt7$ and $0$, so something is wrong.

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