U = 
      [[.6, .8], [.8, -.6]]

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

% Eigenvalues: 12, -13
U*diag([[12, -13]])*U' = 
      [[-4, 12], [12, 3]]

% Eigenvalues: 13, -12
U*diag([[13, -12]])*U' = 
      [[-3, 12], [12, 4]]

% Eigenvalues: 11, -14
U*diag([[11, -14]])*U' = 
      [[-5, 12], [12, 2]]

% Eigenvalues: 14, -11
U*diag([[14, -11]])*U' = 
      [[-2, 12], [12, 5]]

% Eigenvalues: 10, -15
U*diag([[10, -15]])*U' = 
      [[-6, 12], [12, 1]]

% Eigenvalues: 15, -10
U*diag([[15, -10]])*U' = 
      [[-1, 12], [12, 6]]

% Eigenvalues: 9, -16
U*diag([[9, -16]])*U' = 
      [[-7, 12], [12, 0]]

% Eigenvalues: 16, -9
U*diag([[16, -9]])*U' = 
      [[0, 12], [12, 7]]

% Eigenvalues: 17, -8
U*diag([[17, -8]])*U' = 
      [[1, 12], [12, 8]]

% Eigenvalues: 18, -7
U*diag([[18, -7]])*U' = 
      [[2, 12], [12, 9]]

% Eigenvalues: 25, -25
U*diag([[25, -25]])*U' = 
      [[-7, 24], [24, 7]]

% Eigenvalues: 24, -26
U*diag([[24, -26]])*U' = 
      [[-8, 24], [24, 6]]

% Eigenvalues: 26, -24
U*diag([[26, -24]])*U' = 
      [[-6, 24], [24, 8]]

% Eigenvalues: 23, -27
U*diag([[23, -27]])*U' = 
      [[-9, 24], [24, 5]]

% Eigenvalues: 27, -23
U*diag([[27, -23]])*U' = 
      [[-5, 24], [24, 9]]

% Eigenvalues: 22, -28
U*diag([[22, -28]])*U' = 
      [[-10, 24], [24, 4]]

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

% Eigenvalues: 2, -3
U*diag([[20, -30]])*U'/10 = 
      [[-1.2, 2.4], [2.4, .2]]

% Eigenvalues: 3, -2
U*diag([[30, -20]])*U'/10 = 
      [[-.2, 2.4], [2.4, 1.2]]

% Eigenvalues: 1, -4
U*diag([[10, -40]])*U'/10 = 
      [[-2.2, 2.4], [2.4, -.8]]

% Eigenvalues: 4, -1
U*diag([[40, -10]])*U'/10 = 
      [[.8, 2.4], [2.4, 2.2]]

% Eigenvalues: 1, 6
U*diag([[10, 60]])*U'/10 = 
      [[4.2, -2.4], [-2.4, 2.8]]

% Eigenvalues: 5, -5
U*diag([[50, -50]])*U'/10 = 
      [[-1.4, 4.8], [4.8, 1.4]]

% Eigenvalues: 4, -6
U*diag([[40, -60]])*U'/10 = 
      [[-2.4, 4.8], [4.8, .4]]

% Eigenvalues: 6, -4
U*diag([[60, -40]])*U'/10 = 
      [[-.4, 4.8], [4.8, 2.4]]

% Eigenvalues: 7, 2
U*diag([[70, 20]])*U'/10 = 
      [[3.8, 2.4], [2.4, 5.2]]

% Eigenvalues: 3, -7
U*diag([[30, -70]])*U'/10 = 
      [[-3.4, 4.8], [4.8, -.6]]

% Eigenvalues: 7, -3
U*diag([[70, -30]])*U'/10 = 
      [[.6, 4.8], [4.8, 3.4]]

% Eigenvalues: 2, -8
U*diag([[20, -80]])*U'/10 = 
      [[-4.4, 4.8], [4.8, -1.6]]

% Eigenvalues: 8, -2
U*diag([[80, -20]])*U'/10 = 
      [[1.6, 4.8], [4.8, 4.4]]

% Eigenvalues: 8, 3
U*diag([[80, 30]])*U'/10 = 
      [[4.8, 2.4], [2.4, 6.2]]

% Eigenvalues: 1, -9
U*diag([[10, -90]])*U'/10 = 
      [[-5.4, 4.8], [4.8, -2.6]]

% Eigenvalues: 9, 4
U*diag([[90, 40]])*U'/10 = 
      [[5.8, 2.4], [2.4, 7.2]]

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

% Eigenvalues: 1, -1
U*diag([[100, -100]])*U'/100 = 
      [[-.28, .96], [.96, .28]]

% Eigenvalues: 1, -2
U*diag([[100, -200]])*U'/100 = 
      [[-.92, 1.44], [1.44, -.08]]

% Eigenvalues: 1, 2
U*diag([[100, 200]])*U'/100 = 
      [[1.64, -.48], [-.48, 1.36]]

% Eigenvalues: 2, -1
U*diag([[200, -100]])*U'/100 = 
      [[.08, 1.44], [1.44, .92]]

% Eigenvalues: 2, -2
U*diag([[200, -200]])*U'/100 = 
      [[-.56, 1.92], [1.92, .56]]

% Eigenvalues: 1, -3
U*diag([[100, -300]])*U'/100 = 
      [[-1.56, 1.92], [1.92, -.44]]

% Eigenvalues: 1, 3
U*diag([[100, 300]])*U'/100 = 
      [[2.28, -.96], [-.96, 1.72]]

% Eigenvalues: 3, -1
U*diag([[300, -100]])*U'/100 = 
      [[.44, 1.92], [1.92, 1.56]]

% Eigenvalues: 2, 3
U*diag([[200, 300]])*U'/100 = 
      [[2.64, -.48], [-.48, 2.36]]

% Eigenvalues: 1, 4
U*diag([[100, 400]])*U'/100 = 
      [[2.92, -1.44], [-1.44, 2.08]]

% Eigenvalues: 3, -3
U*diag([[300, -300]])*U'/100 = 
      [[-.84, 2.88], [2.88, .84]]

% Eigenvalues: 2, -4
U*diag([[200, -400]])*U'/100 = 
      [[-1.84, 2.88], [2.88, -.16]]

% Eigenvalues: 2, 4
U*diag([[200, 400]])*U'/100 = 
      [[3.28, -.96], [-.96, 2.72]]

% Eigenvalues: 3, -4
U*diag([[300, -400]])*U'/100 = 
      [[-1.48, 3.36], [3.36, .48]]

% Eigenvalues: 3, 4
U*diag([[300, 400]])*U'/100 = 
      [[3.64, -.48], [-.48, 3.36]]

% Eigenvalues: 4, -4
U*diag([[400, -400]])*U'/100 = 
      [[-1.12, 3.84], [3.84, 1.12]]