U = 
      [[1, 0, 0], [0, .6, .28], [0, .8, -.96]]

inv( U ) = 
      [[1, 0, 0], [0, 1.2, .35], [0, 1, -.75]]

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

% Eigenvalues: 1, -50, 50
U*diag([[1, -50, 50]])*inv( U ) = 
      [[1, 0, 0], [0, -22, -21], [0, -96, 22]]

% Eigenvalues: 1, -51, 49
U*diag([[1, -51, 49]])*inv( U ) = 
      [[1, 0, 0], [0, -23, -21], [0, -96, 21]]

% Eigenvalues: 1, -49, 51
U*diag([[1, -49, 51]])*inv( U ) = 
      [[1, 0, 0], [0, -21, -21], [0, -96, 23]]

% Eigenvalues: 2, -50, 50
U*diag([[2, -50, 50]])*inv( U ) = 
      [[2, 0, 0], [0, -22, -21], [0, -96, 22]]

% Eigenvalues: 2, -51, 49
U*diag([[2, -51, 49]])*inv( U ) = 
      [[2, 0, 0], [0, -23, -21], [0, -96, 21]]

% Eigenvalues: 2, -49, 51
U*diag([[2, -49, 51]])*inv( U ) = 
      [[2, 0, 0], [0, -21, -21], [0, -96, 23]]

% Eigenvalues: 1, -52, 48
U*diag([[1, -52, 48]])*inv( U ) = 
      [[1, 0, 0], [0, -24, -21], [0, -96, 20]]

% Eigenvalues: 1, -48, 52
U*diag([[1, -48, 52]])*inv( U ) = 
      [[1, 0, 0], [0, -20, -21], [0, -96, 24]]

% Eigenvalues: 3, -50, 50
U*diag([[3, -50, 50]])*inv( U ) = 
      [[3, 0, 0], [0, -22, -21], [0, -96, 22]]

% Eigenvalues: 3, -51, 49
U*diag([[3, -51, 49]])*inv( U ) = 
      [[3, 0, 0], [0, -23, -21], [0, -96, 21]]

% Eigenvalues: 2, -52, 48
U*diag([[2, -52, 48]])*inv( U ) = 
      [[2, 0, 0], [0, -24, -21], [0, -96, 20]]

% Eigenvalues: 2, -48, 52
U*diag([[2, -48, 52]])*inv( U ) = 
      [[2, 0, 0], [0, -20, -21], [0, -96, 24]]

% Eigenvalues: 1, -53, 47
U*diag([[1, -53, 47]])*inv( U ) = 
      [[1, 0, 0], [0, -25, -21], [0, -96, 19]]

% Eigenvalues: 1, -47, 53
U*diag([[1, -47, 53]])*inv( U ) = 
      [[1, 0, 0], [0, -19, -21], [0, -96, 25]]

% Eigenvalues: 2, -53, 47
U*diag([[2, -53, 47]])*inv( U ) = 
      [[2, 0, 0], [0, -25, -21], [0, -96, 19]]

% Eigenvalues: 1, -69, 31
U*diag([[1, -69, 31]])*inv( U ) = 
      [[1, 0, 0], [0, -41, -21], [0, -96, 3]]

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

% Eigenvalues: 1, -5, 5
U*diag([[10, -50, 50]])*inv( U )/10 = 
      [[1, 0, 0], [0, -2.2, -2.1], [0, -9.6, 2.2]]

% Eigenvalues: 1, -6, 4
U*diag([[10, -60, 40]])*inv( U )/10 = 
      [[1, 0, 0], [0, -3.2, -2.1], [0, -9.6, 1.2]]

% Eigenvalues: 1, -4, 6
U*diag([[10, -40, 60]])*inv( U )/10 = 
      [[1, 0, 0], [0, -1.2, -2.1], [0, -9.6, 3.2]]

% Eigenvalues: 2, -5, 5
U*diag([[20, -50, 50]])*inv( U )/10 = 
      [[2, 0, 0], [0, -2.2, -2.1], [0, -9.6, 2.2]]

% Eigenvalues: 2, -6, 4
U*diag([[20, -60, 40]])*inv( U )/10 = 
      [[2, 0, 0], [0, -3.2, -2.1], [0, -9.6, 1.2]]

% Eigenvalues: 2, -4, 6
U*diag([[20, -40, 60]])*inv( U )/10 = 
      [[2, 0, 0], [0, -1.2, -2.1], [0, -9.6, 3.2]]

% Eigenvalues: 1, -7, 3
U*diag([[10, -70, 30]])*inv( U )/10 = 
      [[1, 0, 0], [0, -4.2, -2.1], [0, -9.6, .2]]

% Eigenvalues: 1, -3, 7
U*diag([[10, -30, 70]])*inv( U )/10 = 
      [[1, 0, 0], [0, -.2, -2.1], [0, -9.6, 4.2]]

% Eigenvalues: 3, -5, 5
U*diag([[30, -50, 50]])*inv( U )/10 = 
      [[3, 0, 0], [0, -2.2, -2.1], [0, -9.6, 2.2]]

% Eigenvalues: 3, -6, 4
U*diag([[30, -60, 40]])*inv( U )/10 = 
      [[3, 0, 0], [0, -3.2, -2.1], [0, -9.6, 1.2]]

% Eigenvalues: 2, -7, 3
U*diag([[20, -70, 30]])*inv( U )/10 = 
      [[2, 0, 0], [0, -4.2, -2.1], [0, -9.6, .2]]

% Eigenvalues: 2, -3, 7
U*diag([[20, -30, 70]])*inv( U )/10 = 
      [[2, 0, 0], [0, -.2, -2.1], [0, -9.6, 4.2]]

% Eigenvalues: 1, -8, 2
U*diag([[10, -80, 20]])*inv( U )/10 = 
      [[1, 0, 0], [0, -5.2, -2.1], [0, -9.6, -.8]]

% Eigenvalues: 1, -2, 8
U*diag([[10, -20, 80]])*inv( U )/10 = 
      [[1, 0, 0], [0, .8, -2.1], [0, -9.6, 5.2]]

% Eigenvalues: 2, -9, 1
U*diag([[20, -90, 10]])*inv( U )/10 = 
      [[2, 0, 0], [0, -6.2, -2.1], [0, -9.6, -1.8]]

% Eigenvalues: 1, -11, -1
U*diag([[10, -110, -10]])*inv( U )/10 = 
      [[1, 0, 0], [0, -8.2, -2.1], [0, -9.6, -3.8]]

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

% Eigenvalues: 1, -2, -1
U*diag([[100, -200, -100]])*inv( U )/100 = 
      [[1, 0, 0], [0, -1.72, -.21], [0, -.96, -1.28]]

% Eigenvalues: 1, -1, 2
U*diag([[100, -100, 200]])*inv( U )/100 = 
      [[1, 0, 0], [0, -.16, -.63], [0, -2.88, 1.16]]

% Eigenvalues: 1, -2, 2
U*diag([[100, -200, 200]])*inv( U )/100 = 
      [[1, 0, 0], [0, -.88, -.84], [0, -3.84, .88]]

% Eigenvalues: 2, -2, -1
U*diag([[200, -200, -100]])*inv( U )/100 = 
      [[2, 0, 0], [0, -1.72, -.21], [0, -.96, -1.28]]

% Eigenvalues: 1, -3, -1
U*diag([[100, -300, -100]])*inv( U )/100 = 
      [[1, 0, 0], [0, -2.44, -.42], [0, -1.92, -1.56]]

% Eigenvalues: 1, -1, 3
U*diag([[100, -100, 300]])*inv( U )/100 = 
      [[1, 0, 0], [0, .12, -.84], [0, -3.84, 1.88]]

% Eigenvalues: 1, -3, -2
U*diag([[100, -300, -200]])*inv( U )/100 = 
      [[1, 0, 0], [0, -2.72, -.21], [0, -.96, -2.28]]

% Eigenvalues: 1, -3, 2
U*diag([[100, -300, 200]])*inv( U )/100 = 
      [[1, 0, 0], [0, -1.6, -1.05], [0, -4.8, .6]]

% Eigenvalues: 1, -2, 3
U*diag([[100, -200, 300]])*inv( U )/100 = 
      [[1, 0, 0], [0, -.6, -1.05], [0, -4.8, 1.6]]

% Eigenvalues: 1, 2, 3
U*diag([[100, 200, 300]])*inv( U )/100 = 
      [[1, 0, 0], [0, 2.28, -.21], [0, -.96, 2.72]]

% Eigenvalues: 2, -3, -1
U*diag([[200, -300, -100]])*inv( U )/100 = 
      [[2, 0, 0], [0, -2.44, -.42], [0, -1.92, -1.56]]

% Eigenvalues: 2, -1, 3
U*diag([[200, -100, 300]])*inv( U )/100 = 
      [[2, 0, 0], [0, .12, -.84], [0, -3.84, 1.88]]

% Eigenvalues: 2, -3, -2
U*diag([[200, -300, -200]])*inv( U )/100 = 
      [[2, 0, 0], [0, -2.72, -.21], [0, -.96, -2.28]]

% Eigenvalues: 2, -2, 3
U*diag([[200, -200, 300]])*inv( U )/100 = 
      [[2, 0, 0], [0, -.6, -1.05], [0, -4.8, 1.6]]

% Eigenvalues: 1, -3, 3
U*diag([[100, -300, 300]])*inv( U )/100 = 
      [[1, 0, 0], [0, -1.32, -1.26], [0, -5.76, 1.32]]

% Eigenvalues: 2, -3, 3
U*diag([[200, -300, 300]])*inv( U )/100 = 
      [[2, 0, 0], [0, -1.32, -1.26], [0, -5.76, 1.32]]