U = 
      [[.2, .4, .4, .8], [.4, -.2, -.8, .4], [.4, .8, -.2, -.4], [.8, -.4, .4, -.2]]

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

% Eigenvalues: 17, -8, -18, 7
U*diag([17, -8, -18, 7])*U' = 
      [[1, 10, -2, 0], [10, -8, 0, 10], [-2, 0, -2, 10], [0, 10, 10, 7]]

% Eigenvalues: 18, -7, -17, 8
U*diag([18, -7, -17, 8])*U' = 
      [[2, 10, -2, 0], [10, -7, 0, 10], [-2, 0, -1, 10], [0, 10, 10, 8]]

% Eigenvalues: 16, -9, -19, 6
U*diag([16, -9, -19, 6])*U' = 
      [[0, 10, -2, 0], [10, -9, 0, 10], [-2, 0, -3, 10], [0, 10, 10, 6]]

% Eigenvalues: 19, -6, -16, 9
U*diag([19, -6, -16, 9])*U' = 
      [[3, 10, -2, 0], [10, -6, 0, 10], [-2, 0, 0, 10], [0, 10, 10, 9]]

% Eigenvalues: 15, -10, -20, 5
U*diag([15, -10, -20, 5])*U' = 
      [[-1, 10, -2, 0], [10, -10, 0, 10], [-2, 0, -4, 10], [0, 10, 10, 5]]

% Eigenvalues: 21, -4, 16, -9
U*diag([21, -4, 16, -9])*U' = 
      [[-3, -6, 2, 8], [-6, 12, 8, 2], [2, 8, 0, 6], [8, 2, 6, 15]]

% Eigenvalues: 22, -3, 17, -8
U*diag([22, -3, 17, -8])*U' = 
      [[-2, -6, 2, 8], [-6, 13, 8, 2], [2, 8, 1, 6], [8, 2, 6, 16]]

% Eigenvalues: 20, -5, -20, 5
U*diag([20, -5, -20, 5])*U' = 
      [[0, 10, 0, 0], [10, -9, 0, 12], [0, 0, 0, 10], [0, 12, 10, 9]]

% Eigenvalues: 19, -6, -21, 4
U*diag([19, -6, -21, 4])*U' = 
      [[-1, 10, 0, 0], [10, -10, 0, 12], [0, 0, -1, 10], [0, 12, 10, 8]]

% Eigenvalues: 21, -4, -19, 6
U*diag([21, -4, -19, 6])*U' = 
      [[1, 10, 0, 0], [10, -8, 0, 12], [0, 0, 1, 10], [0, 12, 10, 10]]

% Eigenvalues: 18, -7, -22, 3
U*diag([18, -7, -22, 3])*U' = 
      [[-2, 10, 0, 0], [10, -11, 0, 12], [0, 0, -2, 10], [0, 12, 10, 7]]

% Eigenvalues: 22, -3, -18, 7
U*diag([22, -3, -18, 7])*U' = 
      [[2, 10, 0, 0], [10, -7, 0, 12], [0, 0, 2, 10], [0, 12, 10, 11]]

% Eigenvalues: 17, -8, -23, 2
U*diag([17, -8, -23, 2])*U' = 
      [[-3, 10, 0, 0], [10, -12, 0, 12], [0, 0, -3, 10], [0, 12, 10, 6]]

% Eigenvalues: 23, -2, 18, -7
U*diag([23, -2, 18, -7])*U' = 
      [[-1, -6, 2, 8], [-6, 14, 8, 2], [2, 8, 2, 6], [8, 2, 6, 17]]

% Eigenvalues: 24, -1, 19, -6
U*diag([24, -1, 19, -6])*U' = 
      [[0, -6, 2, 8], [-6, 15, 8, 2], [2, 8, 3, 6], [8, 2, 6, 18]]

% Eigenvalues: 16, -9, 26, 1
U*diag([16, -9, 26, 1])*U' = 
      [[4, -6, -4, 8], [-6, 19, 8, -4], [-4, 8, -2, 6], [8, -4, 6, 13]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 1 digit %
% beyond the decimal point      %
% with integer eigenvalues      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 3, -2, -3, 2
U*diag([[30, -20, -30, 20]])*U'/10 = 
      [[.6, 2, -.8, 0], [2, -1.2, 0, 1.6], [-.8, 0, -.6, 2], [0, 1.6, 2, 1.2]]

% Eigenvalues: 3, -2, -4, 1
U*diag([[30, -20, -40, 10]])*U'/10 = 
      [[-.2, 2, -.4, 0], [2, -2, 0, 2], [-.4, 0, -.8, 2], [0, 2, 2, 1]]

% Eigenvalues: 3, -2, 4, -1
U*diag([[30, -20, 40, -10]])*U'/10 = 
      [[-.2, -1.2, -.4, 1.6], [-1.2, 2.8, 1.6, -.4], [-.4, 1.6, -.8, 1.2], [1.6, -.4, 1.2, 2.2]]

% Eigenvalues: 4, -1, -3, 2
U*diag([[40, -10, -30, 20]])*U'/10 = 
      [[.8, 2, -.4, 0], [2, -1, 0, 2], [-.4, 0, .2, 2], [0, 2, 2, 2]]

% Eigenvalues: 4, -1, -4, 1
U*diag([[40, -10, -40, 10]])*U'/10 = 
      [[0, 2, 0, 0], [2, -1.8, 0, 2.4], [0, 0, 0, 2], [0, 2.4, 2, 1.8]]

% Eigenvalues: 3, -2, -6, -1
U*diag([[30, -20, -60, -10]])*U'/10 = 
      [[-1.8, 2, .4, 0], [2, -3.6, 0, 2.8], [.4, 0, -1.2, 2], [0, 2.8, 2, .6]]

% Eigenvalues: 3, -2, 6, 1
U*diag([[30, -20, 60, 10]])*U'/10 = 
      [[1.4, -1.2, -1.2, 1.6], [-1.2, 4.4, 1.6, -1.2], [-1.2, 1.6, -.4, 1.2], [1.6, -1.2, 1.2, 2.6]]

% Eigenvalues: 4, -1, 6, 1
U*diag([[40, -10, 60, 10]])*U'/10 = 
      [[1.6, -1.2, -.8, 1.6], [-1.2, 4.6, 1.6, -.8], [-.8, 1.6, .4, 1.2], [1.6, -.8, 1.2, 3.4]]

% Eigenvalues: 3, -2, -5, 5
U*diag([[30, -20, -50, 50]])*U'/10 = 
      [[2.2, 3.6, -1.6, -.8], [3.6, -2, -.8, 2], [-1.6, -.8, -.2, 2.4], [-.8, 2, 2.4, 1]]

% Eigenvalues: 3, -2, -6, 4
U*diag([[30, -20, -60, 40]])*U'/10 = 
      [[1.4, 3.6, -1.2, -.8], [3.6, -2.8, -.8, 2.4], [-1.2, -.8, -.4, 2.4], [-.8, 2.4, 2.4, .8]]

% Eigenvalues: 3, -2, 7, 2
U*diag([[30, -20, 70, 20]])*U'/10 = 
      [[2.2, -1.2, -1.6, 1.6], [-1.2, 5.2, 1.6, -1.6], [-1.6, 1.6, -.2, 1.2], [1.6, -1.6, 1.2, 2.8]]

% Eigenvalues: 4, -1, -7, -2
U*diag([[40, -10, -70, -20]])*U'/10 = 
      [[-2.4, 2, 1.2, 0], [2, -4.2, 0, 3.6], [1.2, 0, -.6, 2], [0, 3.6, 2, 1.2]]

% Eigenvalues: 4, -1, 7, 2
U*diag([[40, -10, 70, 20]])*U'/10 = 
      [[2.4, -1.2, -1.2, 1.6], [-1.2, 5.4, 1.6, -1.2], [-1.2, 1.6, .6, 1.2], [1.6, -1.2, 1.2, 3.6]]

% Eigenvalues: 6, 1, -6, -1
U*diag([[60, 10, -60, -10]])*U'/10 = 
      [[-1.2, 2, 1.6, 0], [2, -3, 0, 4], [1.6, 0, 1.2, 2], [0, 4, 2, 3]]

% Eigenvalues: 7, -3, -4, 1
U*diag([[70, -30, -40, 10]])*U'/10 = 
      [[-.2, 2.4, -.4, .8], [2.4, -1.4, .8, 3.2], [-.4, .8, -.8, 3.6], [.8, 3.2, 3.6, 3.4]]

% Eigenvalues: 4, -1, -7, 3
U*diag([[40, -10, -70, 30]])*U'/10 = 
      [[.8, 3.6, -.4, -.8], [3.6, -3.4, -.8, 3.2], [-.4, -.8, .2, 2.4], [-.8, 3.2, 2.4, 1.4]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matrices with at most 2 digits %
% beyond the decimal point       %
% with integer eigenvalues       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Eigenvalues: 1, -2, -1, 2
U*diag([[100, -200, -100, 200]])*U'/100 = 
      [[.84, 1.2, -1.12, 0], [1.2, -.24, 0, .32], [-1.12, 0, -.84, 1.2], [0, .32, 1.2, .24]]