U = 
      [[2, 2, 1], [2, -1, -2], [1, -2, 2]]/3

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Integer matrices         %
% with integer eigenvalues %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 2, -1, -4
U*diag([[2, -1, -4]])*U' = 
      [[0, 2, 0], [2, -1, 2], [0, 2, -2]]

% Eigenvalues: 1, 4, -2
U*diag([[1, 4, -2]])*U' = 
      [[2, 0, -2], [0, 0, 2], [-2, 2, 1]]

% Eigenvalues: 1, -2, -5
U*diag([[1, -2, -5]])*U' = 
      [[-1, 2, 0], [2, -2, 2], [0, 2, -3]]

% Eigenvalues: 2, 5, -1
U*diag([[2, 5, -1]])*U' = 
      [[3, 0, -2], [0, 1, 2], [-2, 2, 2]]

% Eigenvalues: 4, 7, 1
U*diag([[4, 7, 1]])*U' = 
      [[5, 0, -2], [0, 3, 2], [-2, 2, 4]]

% Eigenvalues: 1, 7, -5
U*diag([[1, 7, -5]])*U' = 
      [[3, 0, -4], [0, -1, 4], [-4, 4, 1]]

% Eigenvalues: 5, -1, -7
U*diag([[5, -1, -7]])*U' = 
      [[1, 4, 0], [4, -1, 4], [0, 4, -3]]

% Eigenvalues: 2, 5, -10
U*diag([[2, 5, -10]])*U' = 
      [[2, 2, -4], [2, -3, 6], [-4, 6, -2]]

% Eigenvalues: 4, 7, -8
U*diag([[4, 7, -8]])*U' = 
      [[4, 2, -4], [2, -1, 6], [-4, 6, 0]]

% Eigenvalues: 8, -4, -7
U*diag([[8, -4, -7]])*U' = 
      [[1, 6, 2], [6, 0, 4], [2, 4, -4]]

% Eigenvalues: 10, -2, -5
U*diag([[10, -2, -5]])*U' = 
      [[3, 6, 2], [6, 2, 4], [2, 4, -2]]

% Eigenvalues: 1, 4, -11
U*diag([[1, 4, -11]])*U' = 
      [[1, 2, -4], [2, -4, 6], [-4, 6, -3]]

% Eigenvalues: 7, -5, -8
U*diag([[7, -5, -8]])*U' = 
      [[0, 6, 2], [6, -1, 4], [2, 4, -5]]

% Eigenvalues: 5, 8, -7
U*diag([[5, 8, -7]])*U' = 
      [[5, 2, -4], [2, 0, 6], [-4, 6, 1]]

% Eigenvalues: 11, -1, -4
U*diag([[11, -1, -4]])*U' = 
      [[4, 6, 2], [6, 3, 4], [2, 4, -1]]

% Eigenvalues: 1, -11, -5
U*diag([[1, -11, -5]])*U' = 
      [[-5, 4, 4], [4, -3, 0], [4, 0, -7]]