U = 
      [[1, 0, .6], [0, .6, .64], [0, .8, -.48]]

inv( U ) = 
      [[1, -.6, .45], [0, .6, .8], [0, 1, -.75]]

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

% Eigenvalues: 5, 10, -15
U*diag([[5, 10, -15]])*inv( U ) = 
      [[5, -12, 9], [0, -6, 12], [0, 12, 1]]

% Eigenvalues: 4, 9, -16
U*diag([[4, 9, -16]])*inv( U ) = 
      [[4, -12, 9], [0, -7, 12], [0, 12, 0]]

% Eigenvalues: 6, 11, -14
U*diag([[6, 11, -14]])*inv( U ) = 
      [[6, -12, 9], [0, -5, 12], [0, 12, 2]]

% Eigenvalues: 3, 8, -17
U*diag([[3, 8, -17]])*inv( U ) = 
      [[3, -12, 9], [0, -8, 12], [0, 12, -1]]

% Eigenvalues: 7, 12, -13
U*diag([[7, 12, -13]])*inv( U ) = 
      [[7, -12, 9], [0, -4, 12], [0, 12, 3]]

% Eigenvalues: 2, 7, -18
U*diag([[2, 7, -18]])*inv( U ) = 
      [[2, -12, 9], [0, -9, 12], [0, 12, -2]]

% Eigenvalues: 8, 13, -12
U*diag([[8, 13, -12]])*inv( U ) = 
      [[8, -12, 9], [0, -3, 12], [0, 12, 4]]

% Eigenvalues: 9, 14, -11
U*diag([[9, 14, -11]])*inv( U ) = 
      [[9, -12, 9], [0, -2, 12], [0, 12, 5]]

% Eigenvalues: 10, 15, -10
U*diag([[10, 15, -10]])*inv( U ) = 
      [[10, -12, 9], [0, -1, 12], [0, 12, 6]]

% Eigenvalues: 11, 16, -9
U*diag([[11, 16, -9]])*inv( U ) = 
      [[11, -12, 9], [0, 0, 12], [0, 12, 7]]

% Eigenvalues: 12, 17, -8
U*diag([[12, 17, -8]])*inv( U ) = 
      [[12, -12, 9], [0, 1, 12], [0, 12, 8]]

% Eigenvalues: 13, 18, -7
U*diag([[13, 18, -7]])*inv( U ) = 
      [[13, -12, 9], [0, 2, 12], [0, 12, 9]]

% Eigenvalues: 14, 19, -6
U*diag([[14, 19, -6]])*inv( U ) = 
      [[14, -12, 9], [0, 3, 12], [0, 12, 10]]

% Eigenvalues: 18, 3, -22
U*diag([[18, 3, -22]])*inv( U ) = 
      [[18, -24, 18], [0, -13, 12], [0, 12, -6]]

% Eigenvalues: 19, 4, -21
U*diag([[19, 4, -21]])*inv( U ) = 
      [[19, -24, 18], [0, -12, 12], [0, 12, -5]]

% Eigenvalues: 17, 2, -23
U*diag([[17, 2, -23]])*inv( U ) = 
      [[17, -24, 18], [0, -14, 12], [0, 12, -7]]

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

% Eigenvalues: 1, -2, 3
U*diag([[10, -20, 30]])*inv( U )/10 = 
      [[1, 1.2, -.9], [0, 1.2, -2.4], [0, -2.4, -.2]]

% Eigenvalues: 1, 2, -3
U*diag([[10, 20, -30]])*inv( U )/10 = 
      [[1, -2.4, 1.8], [0, -1.2, 2.4], [0, 2.4, .2]]

% Eigenvalues: 2, 3, -2
U*diag([[20, 30, -20]])*inv( U )/10 = 
      [[2, -2.4, 1.8], [0, -.2, 2.4], [0, 2.4, 1.2]]

% Eigenvalues: 1, 4, -1
U*diag([[10, 40, -10]])*inv( U )/10 = 
      [[1, -1.2, .9], [0, .8, 2.4], [0, 2.4, 2.2]]

% Eigenvalues: 2, -1, 4
U*diag([[20, -10, 40]])*inv( U )/10 = 
      [[2, 1.2, -.9], [0, 2.2, -2.4], [0, -2.4, .8]]

% Eigenvalues: 2, 1, -4
U*diag([[20, 10, -40]])*inv( U )/10 = 
      [[2, -3.6, 2.7], [0, -2.2, 2.4], [0, 2.4, -.8]]

% Eigenvalues: 3, 2, -3
U*diag([[30, 20, -30]])*inv( U )/10 = 
      [[3, -3.6, 2.7], [0, -1.2, 2.4], [0, 2.4, .2]]

% Eigenvalues: 3, -4, 1
U*diag([[30, -40, 10]])*inv( U )/10 = 
      [[3, -1.2, .9], [0, -.8, -2.4], [0, -2.4, -2.2]]

% Eigenvalues: 3, 4, -1
U*diag([[30, 40, -10]])*inv( U )/10 = 
      [[3, -2.4, 1.8], [0, .8, 2.4], [0, 2.4, 2.2]]

% Eigenvalues: 4, -3, 2
U*diag([[40, -30, 20]])*inv( U )/10 = 
      [[4, -1.2, .9], [0, .2, -2.4], [0, -2.4, -1.2]]

% Eigenvalues: 4, 3, -2
U*diag([[40, 30, -20]])*inv( U )/10 = 
      [[4, -3.6, 2.7], [0, -.2, 2.4], [0, 2.4, 1.2]]

% Eigenvalues: 4, 1, -4
U*diag([[40, 10, -40]])*inv( U )/10 = 
      [[4, -4.8, 3.6], [0, -2.2, 2.4], [0, 2.4, -.8]]

% Eigenvalues: 1, -6, -1
U*diag([[10, -60, -10]])*inv( U )/10 = 
      [[1, -1.2, .9], [0, -2.8, -2.4], [0, -2.4, -4.2]]

% Eigenvalues: 5, -2, 3
U*diag([[50, -20, 30]])*inv( U )/10 = 
      [[5, -1.2, .9], [0, 1.2, -2.4], [0, -2.4, -.2]]

% Eigenvalues: 5, 2, -3
U*diag([[50, 20, -30]])*inv( U )/10 = 
      [[5, -4.8, 3.6], [0, -1.2, 2.4], [0, 2.4, .2]]

% Eigenvalues: 2, 1, 6
U*diag([[20, 10, 60]])*inv( U )/10 = 
      [[2, 2.4, -1.8], [0, 4.2, -2.4], [0, -2.4, 2.8]]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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, -1.2, .9], [0, -1.36, -.48], [0, -.48, -1.64]]

% Eigenvalues: 1, -1, 2
U*diag([[100, -100, 200]])*inv( U )/100 = 
      [[1, .6, -.45], [0, .92, -1.44], [0, -1.44, .08]]

% Eigenvalues: 1, -2, 2
U*diag([[100, -200, 200]])*inv( U )/100 = 
      [[1, .6, -.45], [0, .56, -1.92], [0, -1.92, -.56]]

% Eigenvalues: 2, -2, -1
U*diag([[200, -200, -100]])*inv( U )/100 = 
      [[2, -1.8, 1.35], [0, -1.36, -.48], [0, -.48, -1.64]]

% Eigenvalues: 1, -3, -1
U*diag([[100, -300, -100]])*inv( U )/100 = 
      [[1, -1.2, .9], [0, -1.72, -.96], [0, -.96, -2.28]]

% Eigenvalues: 1, -1, 3
U*diag([[100, -100, 300]])*inv( U )/100 = 
      [[1, 1.2, -.9], [0, 1.56, -1.92], [0, -1.92, .44]]

% Eigenvalues: 1, -3, -2
U*diag([[100, -300, -200]])*inv( U )/100 = 
      [[1, -1.8, 1.35], [0, -2.36, -.48], [0, -.48, -2.64]]

% Eigenvalues: 1, -3, 2
U*diag([[100, -300, 200]])*inv( U )/100 = 
      [[1, .6, -.45], [0, .2, -2.4], [0, -2.4, -1.2]]

% Eigenvalues: 1, 2, 3
U*diag([[100, 200, 300]])*inv( U )/100 = 
      [[1, 1.2, -.9], [0, 2.64, -.48], [0, -.48, 2.36]]

% Eigenvalues: 1, 3, -2
U*diag([[100, 300, -200]])*inv( U )/100 = 
      [[1, -1.8, 1.35], [0, -.2, 2.4], [0, 2.4, 1.2]]

% Eigenvalues: 2, -3, -1
U*diag([[200, -300, -100]])*inv( U )/100 = 
      [[2, -1.8, 1.35], [0, -1.72, -.96], [0, -.96, -2.28]]

% Eigenvalues: 2, -1, 3
U*diag([[200, -100, 300]])*inv( U )/100 = 
      [[2, .6, -.45], [0, 1.56, -1.92], [0, -1.92, .44]]

% Eigenvalues: 3, -2, -1
U*diag([[300, -200, -100]])*inv( U )/100 = 
      [[3, -2.4, 1.8], [0, -1.36, -.48], [0, -.48, -1.64]]

% Eigenvalues: 2, -3, -2
U*diag([[200, -300, -200]])*inv( U )/100 = 
      [[2, -2.4, 1.8], [0, -2.36, -.48], [0, -.48, -2.64]]

% Eigenvalues: 2, -2, 3
U*diag([[200, -200, 300]])*inv( U )/100 = 
      [[2, .6, -.45], [0, 1.2, -2.4], [0, -2.4, -.2]]

% Eigenvalues: 1, -3, 3
U*diag([[100, -300, 300]])*inv( U )/100 = 
      [[1, 1.2, -.9], [0, .84, -2.88], [0, -2.88, -.84]]