U = 
      [[.36, .48, .64], [.48, .64, -.48], [.8, -.6, -.6]]

inv( U ) = 
      [[.84, .12, .8], [.12, .91, -.6], [1, -.75, 0]]

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

% Eigenvalues: 158, 33, -87
U*diag([[158, 33, -87]])*inv( U ) = 
      [[-6, 63, 36], [108, -3, 48], [156, -42, 113]]

% Eigenvalues: 159, 34, -86
U*diag([[159, 34, -86]])*inv( U ) = 
      [[-5, 63, 36], [108, -2, 48], [156, -42, 114]]

% Eigenvalues: 161, 36, -84
U*diag([[161, 36, -84]])*inv( U ) = 
      [[-3, 63, 36], [108, 0, 48], [156, -42, 116]]

% Eigenvalues: 162, 37, -83
U*diag([[162, 37, -83]])*inv( U ) = 
      [[-2, 63, 36], [108, 1, 48], [156, -42, 117]]

% Eigenvalues: 163, 38, -82
U*diag([[163, 38, -82]])*inv( U ) = 
      [[-1, 63, 36], [108, 2, 48], [156, -42, 118]]

% Eigenvalues: 164, 39, -81
U*diag([[164, 39, -81]])*inv( U ) = 
      [[0, 63, 36], [108, 3, 48], [156, -42, 119]]

% Eigenvalues: 166, 41, -79
U*diag([[166, 41, -79]])*inv( U ) = 
      [[2, 63, 36], [108, 5, 48], [156, -42, 121]]

% Eigenvalues: 167, 42, -78
U*diag([[167, 42, -78]])*inv( U ) = 
      [[3, 63, 36], [108, 6, 48], [156, -42, 122]]

% Eigenvalues: 319, 69, -171
U*diag([[319, 69, -171]])*inv( U ) = 
      [[-9, 126, 72], [216, -3, 96], [312, -84, 229]]

% Eigenvalues: 321, 71, -169
U*diag([[321, 71, -169]])*inv( U ) = 
      [[-7, 126, 72], [216, -1, 96], [312, -84, 231]]

% Eigenvalues: 323, 73, -167
U*diag([[323, 73, -167]])*inv( U ) = 
      [[-5, 126, 72], [216, 1, 96], [312, -84, 233]]

% Eigenvalues: 327, 77, -163
U*diag([[327, 77, -163]])*inv( U ) = 
      [[-1, 126, 72], [216, 5, 96], [312, -84, 237]]

% Eigenvalues: 484, 109, -251
U*diag([[484, 109, -251]])*inv( U ) = 
      [[-8, 189, 108], [324, 1, 144], [468, -126, 349]]

% Eigenvalues: 487, 112, -248
U*diag([[487, 112, -248]])*inv( U ) = 
      [[-5, 189, 108], [324, 4, 144], [468, -126, 352]]

% Eigenvalues: 488, 113, -247
U*diag([[488, 113, -247]])*inv( U ) = 
      [[-4, 189, 108], [324, 5, 144], [468, -126, 353]]

% Eigenvalues: 927, 177, -443
U*diag([[927, 177, -443]])*inv( U ) = 
      [[7, 330, 216], [600, -3, 288], [876, -207, 657]]

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

% Eigenvalues: 26, 1, -13
U*diag([[260, 10, -130]])*inv( U )/10 = 
      [[-.4, 7.8, 7.2], [16.8, -2.6, 9.6], [25.2, -3.9, 17]]

% Eigenvalues: 27, 2, -12
U*diag([[270, 20, -120]])*inv( U )/10 = 
      [[.6, 7.8, 7.2], [16.8, -1.6, 9.6], [25.2, -3.9, 18]]

% Eigenvalues: 32, 7, -17
U*diag([[320, 70, -170]])*inv( U )/10 = 
      [[-.8, 12.6, 7.2], [21.6, -.2, 9.6], [31.2, -8.4, 23]]

% Eigenvalues: 33, 8, -16
U*diag([[330, 80, -160]])*inv( U )/10 = 
      [[.2, 12.6, 7.2], [21.6, .8, 9.6], [31.2, -8.4, 24]]

% Eigenvalues: 34, 9, -15
U*diag([[340, 90, -150]])*inv( U )/10 = 
      [[1.2, 12.6, 7.2], [21.6, 1.8, 9.6], [31.2, -8.4, 25]]

% Eigenvalues: 59, 9, -29
U*diag([[590, 90, -290]])*inv( U )/10 = 
      [[-.2, 20.4, 14.4], [38.4, -1.8, 19.2], [56.4, -12.3, 41]]

% Eigenvalues: 65, 15, -33
U*diag([[650, 150, -330]])*inv( U )/10 = 
      [[-.6, 25.2, 14.4], [43.2, .6, 19.2], [62.4, -16.8, 47]]

% Eigenvalues: 67, 17, -31
U*diag([[670, 170, -310]])*inv( U )/10 = 
      [[1.4, 25.2, 14.4], [43.2, 2.6, 19.2], [62.4, -16.8, 49]]

% Eigenvalues: 91, 16, -46
U*diag([[910, 160, -460]])*inv( U )/10 = 
      [[-1, 33, 21.6], [60, -2, 28.8], [87.6, -20.7, 64]]

% Eigenvalues: 92, 17, -45
U*diag([[920, 170, -450]])*inv( U )/10 = 
      [[0, 33, 21.6], [60, -1, 28.8], [87.6, -20.7, 65]]

% Eigenvalues: 98, 23, -49
U*diag([[980, 230, -490]])*inv( U )/10 = 
      [[-.4, 37.8, 21.6], [64.8, 1.4, 28.8], [93.6, -25.2, 71]]

% Eigenvalues: 119, 19, -57
U*diag([[1190, 190, -570]])*inv( U )/10 = 
      [[.6, 40.8, 28.8], [76.8, -2.6, 38.4], [112.8, -24.6, 83]]

% Eigenvalues: 125, 25, -61
U*diag([[1250, 250, -610]])*inv( U )/10 = 
      [[.2, 45.6, 28.8], [81.6, -.2, 38.4], [118.8, -29.1, 89]]

% Eigenvalues: 131, 31, -65
U*diag([[1310, 310, -650]])*inv( U )/10 = 
      [[-.2, 50.4, 28.8], [86.4, 2.2, 38.4], [124.8, -33.6, 95]]

% Eigenvalues: 151, 26, -74
U*diag([[1510, 260, -740]])*inv( U )/10 = 
      [[-.2, 53.4, 36], [98.4, -2.8, 48], [144, -33, 106]]

% Eigenvalues: 157, 32, -78
U*diag([[1570, 320, -780]])*inv( U )/10 = 
      [[-.6, 58.2, 36], [103.2, -.4, 48], [150, -37.5, 112]]

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

% Eigenvalues: 27, 2, -13
U*diag([[2700, 200, -1300]])*inv( U )/100 = 
      [[-.04, 8.28, 7.2], [17.28, -1.96, 9.6], [25.8, -4.35, 18]]

% Eigenvalues: 29, 4, -14
U*diag([[2900, 400, -1400]])*inv( U )/100 = 
      [[.04, 9.72, 7.2], [18.72, -1.04, 9.6], [27.6, -5.7, 20]]

% Eigenvalues: 31, 6, -15
U*diag([[3100, 600, -1500]])*inv( U )/100 = 
      [[.12, 11.16, 7.2], [20.16, -.12, 9.6], [29.4, -7.05, 22]]

% Eigenvalues: 34, 9, -17
U*diag([[3400, 900, -1700]])*inv( U )/100 = 
      [[-.08, 13.56, 7.2], [22.56, 1.08, 9.6], [32.4, -9.3, 25]]

% Eigenvalues: 36, 11, -18
U*diag([[3600, 1100, -1800]])*inv( U )/100 = 
      [[0, 15, 7.2], [24, 2, 9.6], [34.2, -10.65, 27]]

% Eigenvalues: 38, 13, -19
U*diag([[3800, 1300, -1900]])*inv( U )/100 = 
      [[.08, 16.44, 7.2], [25.44, 2.92, 9.6], [36, -12, 29]]

% Eigenvalues: 56, 6, -27
U*diag([[5600, 600, -2700]])*inv( U )/100 = 
      [[0, 18, 14.4], [36, -3, 19.2], [53.4, -10.05, 38]]

% Eigenvalues: 61, 11, -30
U*diag([[6100, 1100, -3000]])*inv( U )/100 = 
      [[-.12, 21.84, 14.4], [39.84, -.88, 19.2], [58.2, -13.65, 43]]

% Eigenvalues: 63, 13, -31
U*diag([[6300, 1300, -3100]])*inv( U )/100 = 
      [[-.04, 23.28, 14.4], [41.28, .04, 19.2], [60, -15, 45]]

% Eigenvalues: 65, 15, -32
U*diag([[6500, 1500, -3200]])*inv( U )/100 = 
      [[.04, 24.72, 14.4], [42.72, .96, 19.2], [61.8, -16.35, 47]]

% Eigenvalues: 67, 17, -33
U*diag([[6700, 1700, -3300]])*inv( U )/100 = 
      [[.12, 26.16, 14.4], [44.16, 1.88, 19.2], [63.6, -17.7, 49]]

% Eigenvalues: 69, 19, -34
U*diag([[6900, 1900, -3400]])*inv( U )/100 = 
      [[.2, 27.6, 14.4], [45.6, 2.8, 19.2], [65.4, -19.05, 51]]

% Eigenvalues: 88, 13, -43
U*diag([[8800, 1300, -4300]])*inv( U )/100 = 
      [[-.16, 30.12, 21.6], [57.12, -2.84, 28.8], [84, -18, 61]]

% Eigenvalues: 90, 15, -44
U*diag([[9000, 1500, -4400]])*inv( U )/100 = 
      [[-.08, 31.56, 21.6], [58.56, -1.92, 28.8], [85.8, -19.35, 63]]

% Eigenvalues: 94, 19, -46
U*diag([[9400, 1900, -4600]])*inv( U )/100 = 
      [[.08, 34.44, 21.6], [61.44, -.08, 28.8], [89.4, -22.05, 67]]

% Eigenvalues: 96, 21, -47
U*diag([[9600, 2100, -4700]])*inv( U )/100 = 
      [[.16, 35.88, 21.6], [62.88, .84, 28.8], [91.2, -23.4, 69]]