Use exactly what Professor Blinder recommends.
A brute force approach would using the Eigenvectors function would be to first write the Mathematica code to generate the matrix m, show the eigenvectors for several values of s, and hope for an epiphany.
s = 3;
diagonal = A + s Table[i (s - i + 1), {i, 2 s + 1}];
offDiagonal = (2 s)^(1/2) \[Epsilon]^(-1) Table[-i, {i, 1, 2 s}];
m = Table[
If[i < j, 0, If[i == j, diagonal[[j]], offDiagonal[[j]]]], {j,
2 s + 1}, {i, 2 s + 1}];
MatrixForm[m]
Eigenvalues[m]
MatrixForm[Eigenvectors[m]]