How to find eigenvectors of an nxn matrix, knowing all the elements?

Questions
1

I have a matrix of size 2s+1 x 2s+1, we’ll call the elements a_ij. All the elements below the diagonal are zero, so it’s upper triangular. Along the diagonal, i.e. the a_ii elements, a_11 = A+is, a_22 = A+i(s-1), a_33 = A+i(s-2) and so on down to a_(2s+1)(2s+1) = A-is. The elements above the diagonal, that is the a_i,i+1 elements are all -i \epsilon^-1 Sqrt[2s]. All other entries are zero. The eigenvalues are obviously the diagonal elements. How do I put this kind of thing into Mathematica and find the general form of the eigenvectors? In other words, there will be 2s+1 eigenvectors, how do I find say the eigenvector for the eigenvalue (s-1) as an example. A and epsilon are constants.

Thanks.

2

Eigenvectors[m]

3

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]]
4

This doesn’t generate the correct matrix. For starters, there are elements in all entries above the diagonal instead of just the ones in “the diagonal above the diagonal”, and the diagonal elements are also wrong. For s=1, the diagonal should be A+i, A, A-i.

5

One of the best features of this forum is the ability to modify code that is provided.

6

Hopefully the modified code below more closely matches the initial description:

s = 3;
diagonal = A + 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]], 
     If[i == j + 1, offDiagonal[[j]], 0]]], {j, 2 s + 1}, {i, 
    2 s + 1}];
MatrixForm[m]
e = Eigenvalues[m]
v = Eigenvectors[m];
MatrixForm[v]

A pattern appears that very likely means a general form can be found.