Eigenvalues of Block Anti-Diagonal Matrix

In line with this answer, I am trying to find the eigenvalues of:

$\mathbf P\mathbf K\mathbf P^\top=\begin{pmatrix}& d_1 & & & & & & \\d_1 & & e_1 & & & & & \\& e_1 & & d_2 & & & & \\& & d_2 & & e_2 & & & \\& & & e_2 & & d_3 & & \\& & & & d_3 & & e_3 & \\& & & & & e_3 & & d_4 \\& & & & & & d_4 & \end{pmatrix}$

which, by the linked post, are the singular values of : $\mathbf B=\begin{pmatrix}d_1&e_1&&\\&d_2&e_2&\\&&d_3&e_3\\&&&d_4\end{pmatrix}$

and $\mathbf K=\left(\begin{array}{c|c}\mathbf 0&\mathbf B^\top \\\hline \mathbf B&\mathbf 0\end{array}\right)$

All the entries of B are real and positive. I do not know the eigenvectors of $\mathbf P\mathbf K\mathbf P^\top$. I am trying to show that the eigenvalues of the matrix (shown below) are natural numbers (disregarding sign).

However, from another incredibly insightful post of J.M.'s, if we can find a Cholesky factorization of the matrix below, then we can put the factorization in block form:

$\mathbf K=\left(\begin{array}{c|c}\mathbf 0&\mathbf B^\top \\\hline \mathbf B&\mathbf 0\end{array}\right)$

And then there exist a similar matrix $\bf H$ to $\bf K$ such that $\bf H$ is closely related to Golub-Kahan tridiagonal matrix (which has known and nice eigenvalues).

My current efforts are focused around trying to find a nice Cholesky factorization of $\mathbf P\mathbf K\mathbf P^\top$.

Any help would be incredibly appreciated!

Edit: I consider even-order matrices. One such example is the matrix:

$\begin{pmatrix} 0 & 3\sqrt{\frac{3}{2}} & 0 &0 \\ 3\sqrt{\frac{3}{2}} & 0 & 3\sqrt{\frac{5}{2}} & 0\\ 0& 3\sqrt{\frac{3}{2}} & 0 & 3 \sqrt{6}\\ 0& 0 & 3 \sqrt{6} &0 \end{pmatrix}$ with eigenvalues $-9,9,-3,3$

so that $\bf{ B} = \begin{pmatrix} 3\sqrt{\frac{3}{2}} & 3\sqrt{\frac{5}{2}}\\ 0 & 3\sqrt{6} \end{pmatrix}$

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What exactly does your bidiagonal matrix look like? – J. M. May 30 '13 at 18:37
I have (n=3)* : $\begin{pmatrix} 0 & \sqrt{5} & 0\\ \sqrt{5} & 0 &\sqrt{10} \\ 0& \sqrt{10} & 0 \end{pmatrix}$ Which has eigenvalues $\mp \sqrt{15}$ and $0$ But I also have several other conjectures of this sort, some with nastier square-roots. I am looking for a more general approach in the general setting (size nxn) – FlamingWilderbeest May 30 '13 at 19:23
Oh, I forgot. I then have that: $\bf B = \begin{pmatrix} \sqrt{5} & \sqrt{10}\\ 0& 0 \end{pmatrix}$ – FlamingWilderbeest May 30 '13 at 19:33
...as I had mentioned in that other answer, the Golub-Kahan shuffle works only for even-order tridiagonals; you need to deflate the additional zero eigenvalue for odd-order matrices before reformulating as a singular value problem. – J. M. May 30 '13 at 19:36
I must have missed that point, although it makes sense as to the necessity of even-orderness. I edited my question to work with the even-order assumption, thanks! – FlamingWilderbeest May 30 '13 at 19:56