U = 
      [[1, .6, .36], [0, .8, .48], [0, 0, .8]]

inv( U ) = 
      [[1, -.75, 0], [0, 1.25, -.75], [0, 0, 1.25]]

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

% Eigenvalues: 1, -11, 9
U*diag([[1, -11, 9]])*inv( U ) = 
      [[1, -9, 9], [0, -11, 12], [0, 0, 9]]

% Eigenvalues: 2, -10, 10
U*diag([[2, -10, 10]])*inv( U ) = 
      [[2, -9, 9], [0, -10, 12], [0, 0, 10]]

% Eigenvalues: 3, -9, 11
U*diag([[3, -9, 11]])*inv( U ) = 
      [[3, -9, 9], [0, -9, 12], [0, 0, 11]]

% Eigenvalues: 1, -7, 13
U*diag([[1, -7, 13]])*inv( U ) = 
      [[1, -6, 9], [0, -7, 12], [0, 0, 13]]

% Eigenvalues: 4, -12, 8
U*diag([[4, -12, 8]])*inv( U ) = 
      [[4, -12, 9], [0, -12, 12], [0, 0, 8]]

% Eigenvalues: 4, -8, 12
U*diag([[4, -8, 12]])*inv( U ) = 
      [[4, -9, 9], [0, -8, 12], [0, 0, 12]]

% Eigenvalues: 3, -13, 7
U*diag([[3, -13, 7]])*inv( U ) = 
      [[3, -12, 9], [0, -13, 12], [0, 0, 7]]

% Eigenvalues: 5, -11, 9
U*diag([[5, -11, 9]])*inv( U ) = 
      [[5, -12, 9], [0, -11, 12], [0, 0, 9]]

% Eigenvalues: 2, -14, 6
U*diag([[2, -14, 6]])*inv( U ) = 
      [[2, -12, 9], [0, -14, 12], [0, 0, 6]]

% Eigenvalues: 6, -10, 10
U*diag([[6, -10, 10]])*inv( U ) = 
      [[6, -12, 9], [0, -10, 12], [0, 0, 10]]

% Eigenvalues: 5, -7, 13
U*diag([[5, -7, 13]])*inv( U ) = 
      [[5, -9, 9], [0, -7, 12], [0, 0, 13]]

% Eigenvalues: 1, -15, 5
U*diag([[1, -15, 5]])*inv( U ) = 
      [[1, -12, 9], [0, -15, 12], [0, 0, 5]]

% Eigenvalues: 7, -9, 11
U*diag([[7, -9, 11]])*inv( U ) = 
      [[7, -12, 9], [0, -9, 12], [0, 0, 11]]

% Eigenvalues: 8, -8, 12
U*diag([[8, -8, 12]])*inv( U ) = 
      [[8, -12, 9], [0, -8, 12], [0, 0, 12]]

% Eigenvalues: 9, -7, 13
U*diag([[9, -7, 13]])*inv( U ) = 
      [[9, -12, 9], [0, -7, 12], [0, 0, 13]]

% Eigenvalues: 1, 17, -3
U*diag([[1, 17, -3]])*inv( U ) = 
      [[1, 12, -9], [0, 17, -12], [0, 0, -3]]

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

% Eigenvalues: 1, 3, -1
U*diag([[10, 30, -10]])*inv( U )/10 = 
      [[1, 1.5, -1.8], [0, 3, -2.4], [0, 0, -1]]

% Eigenvalues: 1, -1, -3
U*diag([[10, -10, -30]])*inv( U )/10 = 
      [[1, -1.5, -.9], [0, -1, -1.2], [0, 0, -3]]

% Eigenvalues: 1, 3, -3
U*diag([[10, 30, -30]])*inv( U )/10 = 
      [[1, 1.5, -2.7], [0, 3, -3.6], [0, 0, -3]]

% Eigenvalues: 3, -1, -3
U*diag([[30, -10, -30]])*inv( U )/10 = 
      [[3, -3, -.9], [0, -1, -1.2], [0, 0, -3]]

% Eigenvalues: 2, 4, -2
U*diag([[20, 40, -20]])*inv( U )/10 = 
      [[2, 1.5, -2.7], [0, 4, -3.6], [0, 0, -2]]

% Eigenvalues: 2, -2, -4
U*diag([[20, -20, -40]])*inv( U )/10 = 
      [[2, -3, -.9], [0, -2, -1.2], [0, 0, -4]]

% Eigenvalues: 1, 5, -1
U*diag([[10, 50, -10]])*inv( U )/10 = 
      [[1, 3, -2.7], [0, 5, -3.6], [0, 0, -1]]

% Eigenvalues: 1, -1, -5
U*diag([[10, -10, -50]])*inv( U )/10 = 
      [[1, -1.5, -1.8], [0, -1, -2.4], [0, 0, -5]]

% Eigenvalues: 1, 3, -5
U*diag([[10, 30, -50]])*inv( U )/10 = 
      [[1, 1.5, -3.6], [0, 3, -4.8], [0, 0, -5]]

% Eigenvalues: 1, 5, -3
U*diag([[10, 50, -30]])*inv( U )/10 = 
      [[1, 3, -3.6], [0, 5, -4.8], [0, 0, -3]]

% Eigenvalues: 3, 5, -1
U*diag([[30, 50, -10]])*inv( U )/10 = 
      [[3, 1.5, -2.7], [0, 5, -3.6], [0, 0, -1]]

% Eigenvalues: 3, -1, -5
U*diag([[30, -10, -50]])*inv( U )/10 = 
      [[3, -3, -1.8], [0, -1, -2.4], [0, 0, -5]]

% Eigenvalues: 3, 5, 1
U*diag([[30, 50, 10]])*inv( U )/10 = 
      [[3, 1.5, -1.8], [0, 5, -2.4], [0, 0, 1]]

% Eigenvalues: 1, -3, -5
U*diag([[10, -30, -50]])*inv( U )/10 = 
      [[1, -3, -.9], [0, -3, -1.2], [0, 0, -5]]

% Eigenvalues: 5, -3, -1
U*diag([[50, -30, -10]])*inv( U )/10 = 
      [[5, -6, .9], [0, -3, 1.2], [0, 0, -1]]

% Eigenvalues: 2, 4, -4
U*diag([[20, 40, -40]])*inv( U )/10 = 
      [[2, 1.5, -3.6], [0, 4, -4.8], [0, 0, -4]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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, -2.25, .45], [0, -2, .6], [0, 0, -1]]

% Eigenvalues: 1, -1, 2
U*diag([[100, -100, 200]])*inv( U )/100 = 
      [[1, -1.5, 1.35], [0, -1, 1.8], [0, 0, 2]]

% Eigenvalues: 1, -2, 2
U*diag([[100, -200, 200]])*inv( U )/100 = 
      [[1, -2.25, 1.8], [0, -2, 2.4], [0, 0, 2]]

% Eigenvalues: 2, -2, -1
U*diag([[200, -200, -100]])*inv( U )/100 = 
      [[2, -3, .45], [0, -2, .6], [0, 0, -1]]

% Eigenvalues: 1, -3, -2
U*diag([[100, -300, -200]])*inv( U )/100 = 
      [[1, -3, .45], [0, -3, .6], [0, 0, -2]]

% Eigenvalues: 1, -3, 2
U*diag([[100, -300, 200]])*inv( U )/100 = 
      [[1, -3, 2.25], [0, -3, 3], [0, 0, 2]]

% Eigenvalues: 1, -2, 3
U*diag([[100, -200, 300]])*inv( U )/100 = 
      [[1, -2.25, 2.25], [0, -2, 3], [0, 0, 3]]

% Eigenvalues: 1, 2, 3
U*diag([[100, 200, 300]])*inv( U )/100 = 
      [[1, .75, .45], [0, 2, .6], [0, 0, 3]]

% Eigenvalues: 2, -3, -1
U*diag([[200, -300, -100]])*inv( U )/100 = 
      [[2, -3.75, .9], [0, -3, 1.2], [0, 0, -1]]

% Eigenvalues: 2, -1, 3
U*diag([[200, -100, 300]])*inv( U )/100 = 
      [[2, -2.25, 1.8], [0, -1, 2.4], [0, 0, 3]]

% Eigenvalues: 3, -2, -1
U*diag([[300, -200, -100]])*inv( U )/100 = 
      [[3, -3.75, .45], [0, -2, .6], [0, 0, -1]]

% Eigenvalues: 2, -3, -2
U*diag([[200, -300, -200]])*inv( U )/100 = 
      [[2, -3.75, .45], [0, -3, .6], [0, 0, -2]]

% Eigenvalues: 2, -2, 3
U*diag([[200, -200, 300]])*inv( U )/100 = 
      [[2, -3, 2.25], [0, -2, 3], [0, 0, 3]]

% Eigenvalues: 2, -3, 3
U*diag([[200, -300, 300]])*inv( U )/100 = 
      [[2, -3.75, 2.7], [0, -3, 3.6], [0, 0, 3]]

% Eigenvalues: 3, -3, -2
U*diag([[300, -300, -200]])*inv( U )/100 = 
      [[3, -4.5, .45], [0, -3, .6], [0, 0, -2]]