I'm trying to generate the characteristic polynomial of the matrix below, from the reduced matrix. But the calculations are not correct, because, probably, my reduced matrix is wrong. I hope this community helps.
Reduced matrix:
(\begin{array}{rrrr} 0 & \, \sqrt{\frac{1}{2}} & \, \sqrt{\frac{1}{2}} & \, \sqrt{n - \frac{2}{n - 1}} \\ \sqrt{\frac{1}{2}} & 0 & \sqrt{\frac{1}{2}} & 0 \\ \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & 0 & 0 \\ \, \sqrt{n - \frac{2}{n - 1}} & 0 & 0 & 0 \end{array}
I got $$ \left( \; \; x^4 -\left( \frac{3}{2} + \frac{n^2-5n+6}{n-1}\right) \,x^2 - \,\frac{1}{\sqrt 2} \, x + \left( \frac{n^2-5n+6}{n-1} \right) \; \right) \; x^{n-4} $$
This assumes that the matrix is $n$ by $n$ which is my best guess. Note that most of the eigenvalues are zero; it is symmetric so all are real.