U = [[.6, .48, .64], [.8, -.36, -.48], [0, -.8, .6]] inv( U ) = [[.6, .8, 0], [.48, -.36, -.8], [.64, -.48, .6]] %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Integer matrices % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 5, 60, -65 U*diag([5, 60, -65])*inv( U ) = [[-11, 12, -48], [12, -4, 36], [-48, 36, 15]] % Eigenvalues: 4, 59, -66 U*diag([4, 59, -66])*inv( U ) = [[-12, 12, -48], [12, -5, 36], [-48, 36, 14]] % Eigenvalues: 6, 61, -64 U*diag([6, 61, -64])*inv( U ) = [[-10, 12, -48], [12, -3, 36], [-48, 36, 16]] % Eigenvalues: 3, 58, -67 U*diag([3, 58, -67])*inv( U ) = [[-13, 12, -48], [12, -6, 36], [-48, 36, 13]] % Eigenvalues: 7, 62, -63 U*diag([7, 62, -63])*inv( U ) = [[-9, 12, -48], [12, -2, 36], [-48, 36, 17]] % Eigenvalues: 2, 57, -68 U*diag([2, 57, -68])*inv( U ) = [[-14, 12, -48], [12, -7, 36], [-48, 36, 12]] % Eigenvalues: 8, 63, -62 U*diag([8, 63, -62])*inv( U ) = [[-8, 12, -48], [12, -1, 36], [-48, 36, 18]] % Eigenvalues: 12, -68, 57 U*diag([12, -68, 57])*inv( U ) = [[12, 0, 48], [0, 12, -36], [48, -36, -23]] % Eigenvalues: 11, -69, 56 U*diag([11, -69, 56])*inv( U ) = [[11, 0, 48], [0, 11, -36], [48, -36, -24]] % Eigenvalues: 13, -67, 58 U*diag([13, -67, 58])*inv( U ) = [[13, 0, 48], [0, 13, -36], [48, -36, -22]] % Eigenvalues: 10, -70, 55 U*diag([10, -70, 55])*inv( U ) = [[10, 0, 48], [0, 10, -36], [48, -36, -25]] % Eigenvalues: 14, -66, 59 U*diag([14, -66, 59])*inv( U ) = [[14, 0, 48], [0, 14, -36], [48, -36, -21]] % Eigenvalues: 9, -71, 54 U*diag([9, -71, 54])*inv( U ) = [[9, 0, 48], [0, 9, -36], [48, -36, -26]] % Eigenvalues: 13, 68, -57 U*diag([13, 68, -57])*inv( U ) = [[-3, 12, -48], [12, 4, 36], [-48, 36, 23]] % Eigenvalues: 14, 69, -56 U*diag([14, 69, -56])*inv( U ) = [[-2, 12, -48], [12, 5, 36], [-48, 36, 24]] % Eigenvalues: 15, 70, -55 U*diag([15, 70, -55])*inv( U ) = [[-1, 12, -48], [12, 6, 36], [-48, 36, 25]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 1 digit % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, 12, -13 U*diag([[10, 120, -130]])*inv( U )/10 = [[-2.2, 2.4, -9.6], [2.4, -.8, 7.2], [-9.6, 7.2, 3]] % Eigenvalues: 2, 13, -12 U*diag([[20, 130, -120]])*inv( U )/10 = [[-1.2, 2.4, -9.6], [2.4, .2, 7.2], [-9.6, 7.2, 4]] % Eigenvalues: 2, -14, 11 U*diag([[20, -140, 110]])*inv( U )/10 = [[2, 0, 9.6], [0, 2, -7.2], [9.6, -7.2, -5]] % Eigenvalues: 3, -13, 12 U*diag([[30, -130, 120]])*inv( U )/10 = [[3, 0, 9.6], [0, 3, -7.2], [9.6, -7.2, -4]] % Eigenvalues: 1, -10, 15 U*diag([[10, -100, 150]])*inv( U )/10 = [[4.2, -2.4, 9.6], [-2.4, 2.8, -7.2], [9.6, -7.2, -1]] % Eigenvalues: 1, -15, 10 U*diag([[10, -150, 100]])*inv( U )/10 = [[1, 0, 9.6], [0, 1, -7.2], [9.6, -7.2, -6]] % Eigenvalues: 3, 14, -11 U*diag([[30, 140, -110]])*inv( U )/10 = [[-.2, 2.4, -9.6], [2.4, 1.2, 7.2], [-9.6, 7.2, 5]] % Eigenvalues: 4, -12, 13 U*diag([[40, -120, 130]])*inv( U )/10 = [[4, 0, 9.6], [0, 4, -7.2], [9.6, -7.2, -3]] % Eigenvalues: 2, -9, 16 U*diag([[20, -90, 160]])*inv( U )/10 = [[5.2, -2.4, 9.6], [-2.4, 3.8, -7.2], [9.6, -7.2, 0]] % Eigenvalues: 4, 15, -10 U*diag([[40, 150, -100]])*inv( U )/10 = [[.8, 2.4, -9.6], [2.4, 2.2, 7.2], [-9.6, 7.2, 6]] % Eigenvalues: 4, 10, -15 U*diag([[40, 100, -150]])*inv( U )/10 = [[-2.4, 4.8, -9.6], [4.8, .4, 7.2], [-9.6, 7.2, 1]] % Eigenvalues: 5, -11, 14 U*diag([[50, -110, 140]])*inv( U )/10 = [[5, 0, 9.6], [0, 5, -7.2], [9.6, -7.2, -2]] % Eigenvalues: 5, 11, -14 U*diag([[50, 110, -140]])*inv( U )/10 = [[-1.4, 4.8, -9.6], [4.8, 1.4, 7.2], [-9.6, 7.2, 2]] % Eigenvalues: 3, 9, -16 U*diag([[30, 90, -160]])*inv( U )/10 = [[-3.4, 4.8, -9.6], [4.8, -.6, 7.2], [-9.6, 7.2, 0]] % Eigenvalues: 6, 12, -13 U*diag([[60, 120, -130]])*inv( U )/10 = [[-.4, 4.8, -9.6], [4.8, 2.4, 7.2], [-9.6, 7.2, 3]] % Eigenvalues: 2, 8, -17 U*diag([[20, 80, -170]])*inv( U )/10 = [[-4.4, 4.8, -9.6], [4.8, -1.6, 7.2], [-9.6, 7.2, -1]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 2 digits % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, -13, 12 U*diag([[100, -1300, 1200]])*inv( U )/100 = [[2.28, -.96, 9.6], [-.96, 1.72, -7.2], [9.6, -7.2, -4]] % Eigenvalues: 1, -12, 13 U*diag([[100, -1200, 1300]])*inv( U )/100 = [[2.92, -1.44, 9.6], [-1.44, 2.08, -7.2], [9.6, -7.2, -3]] % Eigenvalues: 2, -13, 12 U*diag([[200, -1300, 1200]])*inv( U )/100 = [[2.64, -.48, 9.6], [-.48, 2.36, -7.2], [9.6, -7.2, -4]] % Eigenvalues: 2, -12, 13 U*diag([[200, -1200, 1300]])*inv( U )/100 = [[3.28, -.96, 9.6], [-.96, 2.72, -7.2], [9.6, -7.2, -3]] % Eigenvalues: 1, -14, 11 U*diag([[100, -1400, 1100]])*inv( U )/100 = [[1.64, -.48, 9.6], [-.48, 1.36, -7.2], [9.6, -7.2, -5]] % Eigenvalues: 1, -11, 14 U*diag([[100, -1100, 1400]])*inv( U )/100 = [[3.56, -1.92, 9.6], [-1.92, 2.44, -7.2], [9.6, -7.2, -2]] % Eigenvalues: 2, -11, 14 U*diag([[200, -1100, 1400]])*inv( U )/100 = [[3.92, -1.44, 9.6], [-1.44, 3.08, -7.2], [9.6, -7.2, -2]] % Eigenvalues: 2, 11, -14 U*diag([[200, 1100, -1400]])*inv( U )/100 = [[-2.48, 3.36, -9.6], [3.36, -.52, 7.2], [-9.6, 7.2, 2]] % Eigenvalues: 3, -12, 13 U*diag([[300, -1200, 1300]])*inv( U )/100 = [[3.64, -.48, 9.6], [-.48, 3.36, -7.2], [9.6, -7.2, -3]] % Eigenvalues: 3, 12, -13 U*diag([[300, 1200, -1300]])*inv( U )/100 = [[-1.48, 3.36, -9.6], [3.36, .48, 7.2], [-9.6, 7.2, 3]] % Eigenvalues: 1, 10, -15 U*diag([[100, 1000, -1500]])*inv( U )/100 = [[-3.48, 3.36, -9.6], [3.36, -1.52, 7.2], [-9.6, 7.2, 1]] % Eigenvalues: 1, 15, -10 U*diag([[100, 1500, -1000]])*inv( U )/100 = [[-.28, .96, -9.6], [.96, .28, 7.2], [-9.6, 7.2, 6]] % Eigenvalues: 3, -14, 11 U*diag([[300, -1400, 1100]])*inv( U )/100 = [[2.36, .48, 9.6], [.48, 2.64, -7.2], [9.6, -7.2, -5]] % Eigenvalues: 3, -11, 14 U*diag([[300, -1100, 1400]])*inv( U )/100 = [[4.28, -.96, 9.6], [-.96, 3.72, -7.2], [9.6, -7.2, -2]] % Eigenvalues: 2, -15, 10 U*diag([[200, -1500, 1000]])*inv( U )/100 = [[1.36, .48, 9.6], [.48, 1.64, -7.2], [9.6, -7.2, -6]] % Eigenvalues: 2, -10, 15 U*diag([[200, -1000, 1500]])*inv( U )/100 = [[4.56, -1.92, 9.6], [-1.92, 3.44, -7.2], [9.6, -7.2, -1]]