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

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

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

% Eigenvalues: 2, -2
U*diag([[2, -2]])*inv( U ) = 
      [[2, -3], [0, -2]]

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

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

% Eigenvalues: 5, 1
U*diag([[5, 1]])*inv( U ) = 
      [[5, -3], [0, 1]]

% Eigenvalues: 4, -4
U*diag([[4, -4]])*inv( U ) = 
      [[4, -6], [0, -4]]

% Eigenvalues: 3, -5
U*diag([[3, -5]])*inv( U ) = 
      [[3, -6], [0, -5]]

% Eigenvalues: 5, -3
U*diag([[5, -3]])*inv( U ) = 
      [[5, -6], [0, -3]]

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

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

% Eigenvalues: 6, 2
U*diag([[6, 2]])*inv( U ) = 
      [[6, -3], [0, 2]]

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

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

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

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

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

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

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

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

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

% Eigenvalues: 3, -3
U*diag([[30, -30]])*inv( U )/10 = 
      [[3, -4.5], [0, -3]]

% Eigenvalues: 2, -4
U*diag([[20, -40]])*inv( U )/10 = 
      [[2, -4.5], [0, -4]]

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

% Eigenvalues: 4, -2
U*diag([[40, -20]])*inv( U )/10 = 
      [[4, -4.5], [0, -2]]

% Eigenvalues: 1, -5
U*diag([[10, -50]])*inv( U )/10 = 
      [[1, -4.5], [0, -5]]

% Eigenvalues: 5, -1
U*diag([[50, -10]])*inv( U )/10 = 
      [[5, -4.5], [0, -1]]

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

% Eigenvalues: 1, 7
U*diag([[10, 70]])*inv( U )/10 = 
      [[1, 4.5], [0, 7]]

% Eigenvalues: 4, -6
U*diag([[40, -60]])*inv( U )/10 = 
      [[4, -7.5], [0, -6]]

% Eigenvalues: 4, 6
U*diag([[40, 60]])*inv( U )/10 = 
      [[4, 1.5], [0, 6]]

% Eigenvalues: 3, -7
U*diag([[30, -70]])*inv( U )/10 = 
      [[3, -7.5], [0, -7]]

% Eigenvalues: 2, -8
U*diag([[20, -80]])*inv( U )/10 = 
      [[2, -7.5], [0, -8]]

% Eigenvalues: 2, 8
U*diag([[20, 80]])*inv( U )/10 = 
      [[2, 4.5], [0, 8]]

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

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

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

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

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

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

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

% Eigenvalues: 1, -4
U*diag([[100, -400]])*inv( U )/100 = 
      [[1, -3.75], [0, -4]]

% Eigenvalues: 1, 4
U*diag([[100, 400]])*inv( U )/100 = 
      [[1, 2.25], [0, 4]]

% Eigenvalues: 4, -1
U*diag([[400, -100]])*inv( U )/100 = 
      [[4, -3.75], [0, -1]]

% Eigenvalues: 3, -4
U*diag([[300, -400]])*inv( U )/100 = 
      [[3, -5.25], [0, -4]]

% Eigenvalues: 3, 4
U*diag([[300, 400]])*inv( U )/100 = 
      [[3, .75], [0, 4]]

% Eigenvalues: 4, -3
U*diag([[400, -300]])*inv( U )/100 = 
      [[4, -5.25], [0, -3]]