U = [[.28, .36, .48], [.96, -.48, -.64], [0, -.8, .6]] inv( U ) = [[1, .75, 0], [.72, -.21, -.8], [.96, -.28, .6]] %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Integer matrices % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 12, -68, 57 U*diag([[12, -68, 57]])*inv( U ) = [[12, 0, 36], [0, 12, -48], [72, -21, -23]] % Eigenvalues: 11, -69, 56 U*diag([[11, -69, 56]])*inv( U ) = [[11, 0, 36], [0, 11, -48], [72, -21, -24]] % Eigenvalues: 13, -67, 58 U*diag([[13, -67, 58]])*inv( U ) = [[13, 0, 36], [0, 13, -48], [72, -21, -22]] % Eigenvalues: 14, -66, 59 U*diag([[14, -66, 59]])*inv( U ) = [[14, 0, 36], [0, 14, -48], [72, -21, -21]] % Eigenvalues: 9, -71, 54 U*diag([[9, -71, 54]])*inv( U ) = [[9, 0, 36], [0, 9, -48], [72, -21, -26]] % Eigenvalues: 16, -64, 61 U*diag([[16, -64, 61]])*inv( U ) = [[16, 0, 36], [0, 16, -48], [72, -21, -19]] % Eigenvalues: 17, -63, 62 U*diag([[17, -63, 62]])*inv( U ) = [[17, 0, 36], [0, 17, -48], [72, -21, -18]] % Eigenvalues: 18, -62, 63 U*diag([[18, -62, 63]])*inv( U ) = [[18, 0, 36], [0, 18, -48], [72, -21, -17]] % Eigenvalues: 19, -61, 64 U*diag([[19, -61, 64]])*inv( U ) = [[19, 0, 36], [0, 19, -48], [72, -21, -16]] % Eigenvalues: 32, -48, 77 U*diag([[32, -48, 77]])*inv( U ) = [[32, 0, 36], [0, 32, -48], [72, -21, -3]] % Eigenvalues: 33, -47, 78 U*diag([[33, -47, 78]])*inv( U ) = [[33, 0, 36], [0, 33, -48], [72, -21, -2]] % Eigenvalues: 34, -46, 79 U*diag([[34, -46, 79]])*inv( U ) = [[34, 0, 36], [0, 34, -48], [72, -21, -1]] % Eigenvalues: 36, -44, 81 U*diag([[36, -44, 81]])*inv( U ) = [[36, 0, 36], [0, 36, -48], [72, -21, 1]] % Eigenvalues: 37, -43, 82 U*diag([[37, -43, 82]])*inv( U ) = [[37, 0, 36], [0, 37, -48], [72, -21, 2]] % Eigenvalues: 38, -42, 83 U*diag([[38, -42, 83]])*inv( U ) = [[38, 0, 36], [0, 38, -48], [72, -21, 3]] % Eigenvalues: 54, 34, -91 U*diag([[54, 34, -91]])*inv( U ) = [[-18, 21, -36], [96, 26, 48], [-72, 21, -11]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 1 digit % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 2, -14, 11 U*diag([[20, -140, 110]])*inv( U )/10 = [[2, 0, 7.2], [0, 2, -9.6], [14.4, -4.2, -5]] % Eigenvalues: 3, -13, 12 U*diag([[30, -130, 120]])*inv( U )/10 = [[3, 0, 7.2], [0, 3, -9.6], [14.4, -4.2, -4]] % Eigenvalues: 1, -15, 10 U*diag([[10, -150, 100]])*inv( U )/10 = [[1, 0, 7.2], [0, 1, -9.6], [14.4, -4.2, -6]] % Eigenvalues: 4, -12, 13 U*diag([[40, -120, 130]])*inv( U )/10 = [[4, 0, 7.2], [0, 4, -9.6], [14.4, -4.2, -3]] % Eigenvalues: 4, 10, -15 U*diag([[40, 100, -150]])*inv( U )/10 = [[-3.2, 2.1, -7.2], [9.6, 1.2, 9.6], [-14.4, 4.2, 1]] % Eigenvalues: 5, -11, 14 U*diag([[50, -110, 140]])*inv( U )/10 = [[5, 0, 7.2], [0, 5, -9.6], [14.4, -4.2, -2]] % Eigenvalues: 5, 11, -14 U*diag([[50, 110, -140]])*inv( U )/10 = [[-2.2, 2.1, -7.2], [9.6, 2.2, 9.6], [-14.4, 4.2, 2]] % Eigenvalues: 3, 9, -16 U*diag([[30, 90, -160]])*inv( U )/10 = [[-4.2, 2.1, -7.2], [9.6, .2, 9.6], [-14.4, 4.2, 0]] % Eigenvalues: 6, 12, -13 U*diag([[60, 120, -130]])*inv( U )/10 = [[-1.2, 2.1, -7.2], [9.6, 3.2, 9.6], [-14.4, 4.2, 3]] % Eigenvalues: 2, 8, -17 U*diag([[20, 80, -170]])*inv( U )/10 = [[-5.2, 2.1, -7.2], [9.6, -.8, 9.6], [-14.4, 4.2, -1]] % Eigenvalues: 6, -10, 15 U*diag([[60, -100, 150]])*inv( U )/10 = [[6, 0, 7.2], [0, 6, -9.6], [14.4, -4.2, -1]] % Eigenvalues: 7, 13, -12 U*diag([[70, 130, -120]])*inv( U )/10 = [[-.2, 2.1, -7.2], [9.6, 4.2, 9.6], [-14.4, 4.2, 4]] % Eigenvalues: 8, 14, -11 U*diag([[80, 140, -110]])*inv( U )/10 = [[.8, 2.1, -7.2], [9.6, 5.2, 9.6], [-14.4, 4.2, 5]] % Eigenvalues: 9, 15, -10 U*diag([[90, 150, -100]])*inv( U )/10 = [[1.8, 2.1, -7.2], [9.6, 6.2, 9.6], [-14.4, 4.2, 6]] % Eigenvalues: 9, -17, 8 U*diag([[90, -170, 80]])*inv( U )/10 = [[1.8, 2.1, 7.2], [9.6, 6.2, -9.6], [14.4, -4.2, -8]] % Eigenvalues: 8, -18, 7 U*diag([[80, -180, 70]])*inv( U )/10 = [[.8, 2.1, 7.2], [9.6, 5.2, -9.6], [14.4, -4.2, -9]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 2 digits % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, -12, 13 U*diag([[100, -1200, 1300]])*inv( U )/100 = [[3.16, -.63, 7.2], [-2.88, 1.84, -9.6], [14.4, -4.2, -3]] % Eigenvalues: 1, -13, 12 U*diag([[100, -1300, 1200]])*inv( U )/100 = [[2.44, -.42, 7.2], [-1.92, 1.56, -9.6], [14.4, -4.2, -4]] % Eigenvalues: 2, -12, 13 U*diag([[200, -1200, 1300]])*inv( U )/100 = [[3.44, -.42, 7.2], [-1.92, 2.56, -9.6], [14.4, -4.2, -3]] % Eigenvalues: 2, -13, 12 U*diag([[200, -1300, 1200]])*inv( U )/100 = [[2.72, -.21, 7.2], [-.96, 2.28, -9.6], [14.4, -4.2, -4]] % Eigenvalues: 1, -14, 11 U*diag([[100, -1400, 1100]])*inv( U )/100 = [[1.72, -.21, 7.2], [-.96, 1.28, -9.6], [14.4, -4.2, -5]] % Eigenvalues: 1, 14, -11 U*diag([[100, 1400, -1100]])*inv( U )/100 = [[-1.16, .63, -7.2], [2.88, .16, 9.6], [-14.4, 4.2, 5]] % Eigenvalues: 2, -11, 14 U*diag([[200, -1100, 1400]])*inv( U )/100 = [[4.16, -.63, 7.2], [-2.88, 2.84, -9.6], [14.4, -4.2, -2]] % Eigenvalues: 3, -12, 13 U*diag([[300, -1200, 1300]])*inv( U )/100 = [[3.72, -.21, 7.2], [-.96, 3.28, -9.6], [14.4, -4.2, -3]] % Eigenvalues: 3, -11, 14 U*diag([[300, -1100, 1400]])*inv( U )/100 = [[4.44, -.42, 7.2], [-1.92, 3.56, -9.6], [14.4, -4.2, -2]] % Eigenvalues: 3, -14, 11 U*diag([[300, -1400, 1100]])*inv( U )/100 = [[2.28, .21, 7.2], [.96, 2.72, -9.6], [14.4, -4.2, -5]] % Eigenvalues: 1, 15, -10 U*diag([[100, 1500, -1000]])*inv( U )/100 = [[-.44, .42, -7.2], [1.92, .44, 9.6], [-14.4, 4.2, 6]] % Eigenvalues: 2, -15, 10 U*diag([[200, -1500, 1000]])*inv( U )/100 = [[1.28, .21, 7.2], [.96, 1.72, -9.6], [14.4, -4.2, -6]] % Eigenvalues: 4, -13, 12 U*diag([[400, -1300, 1200]])*inv( U )/100 = [[3.28, .21, 7.2], [.96, 3.72, -9.6], [14.4, -4.2, -4]] % Eigenvalues: 2, 15, -10 U*diag([[200, 1500, -1000]])*inv( U )/100 = [[-.16, .63, -7.2], [2.88, 1.16, 9.6], [-14.4, 4.2, 6]] % Eigenvalues: 4, -11, 14 U*diag([[400, -1100, 1400]])*inv( U )/100 = [[4.72, -.21, 7.2], [-.96, 4.28, -9.6], [14.4, -4.2, -2]] % Eigenvalues: 4, -14, 11 U*diag([[400, -1400, 1100]])*inv( U )/100 = [[2.56, .42, 7.2], [1.92, 3.44, -9.6], [14.4, -4.2, -5]]