U = [[1, .36, .48], [0, .48, .64], [0, .8, -.6]] inv( U ) = [[1, -.75, 0], [0, .75, .8], [0, 1, -.6]] %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Integer matrices % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, -63, 62 U*diag([[1, -63, 62]])*inv( U ) = [[1, 12, -36], [0, 17, -48], [0, -75, -18]] % Eigenvalues: 2, 62, -63 U*diag([[2, 62, -63]])*inv( U ) = [[2, -15, 36], [0, -18, 48], [0, 75, 17]] % Eigenvalues: 2, -62, 63 U*diag([[2, -62, 63]])*inv( U ) = [[2, 12, -36], [0, 18, -48], [0, -75, -17]] % Eigenvalues: 1, 61, -64 U*diag([[1, 61, -64]])*inv( U ) = [[1, -15, 36], [0, -19, 48], [0, 75, 16]] % Eigenvalues: 3, 63, -62 U*diag([[3, 63, -62]])*inv( U ) = [[3, -15, 36], [0, -17, 48], [0, 75, 18]] % Eigenvalues: 1, 65, -60 U*diag([[1, 65, -60]])*inv( U ) = [[1, -12, 36], [0, -15, 48], [0, 75, 20]] % Eigenvalues: 3, -61, 64 U*diag([[3, -61, 64]])*inv( U ) = [[3, 12, -36], [0, 19, -48], [0, -75, -16]] % Eigenvalues: 4, 64, -61 U*diag([[4, 64, -61]])*inv( U ) = [[4, -15, 36], [0, -16, 48], [0, 75, 19]] % Eigenvalues: 4, -64, 61 U*diag([[4, -64, 61]])*inv( U ) = [[4, 9, -36], [0, 16, -48], [0, -75, -19]] % Eigenvalues: 3, -65, 60 U*diag([[3, -65, 60]])*inv( U ) = [[3, 9, -36], [0, 15, -48], [0, -75, -20]] % Eigenvalues: 1, -59, 66 U*diag([[1, -59, 66]])*inv( U ) = [[1, 15, -36], [0, 21, -48], [0, -75, -14]] % Eigenvalues: 2, 66, -59 U*diag([[2, 66, -59]])*inv( U ) = [[2, -12, 36], [0, -14, 48], [0, 75, 21]] % Eigenvalues: 2, -66, 59 U*diag([[2, -66, 59]])*inv( U ) = [[2, 9, -36], [0, 14, -48], [0, -75, -21]] % Eigenvalues: 3, 59, -66 U*diag([[3, 59, -66]])*inv( U ) = [[3, -18, 36], [0, -21, 48], [0, 75, 14]] % Eigenvalues: 1, -67, 58 U*diag([[1, -67, 58]])*inv( U ) = [[1, 9, -36], [0, 13, -48], [0, -75, -22]] % Eigenvalues: 2, 58, -67 U*diag([[2, 58, -67]])*inv( U ) = [[2, -18, 36], [0, -22, 48], [0, 75, 13]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 1 digit % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, -13, 12 U*diag([[10, -130, 120]])*inv( U )/10 = [[1, 1.5, -7.2], [0, 3, -9.6], [0, -15, -4]] % Eigenvalues: 1, 13, -12 U*diag([[10, 130, -120]])*inv( U )/10 = [[1, -3, 7.2], [0, -3, 9.6], [0, 15, 4]] % Eigenvalues: 2, -12, 13 U*diag([[20, -120, 130]])*inv( U )/10 = [[2, 1.5, -7.2], [0, 4, -9.6], [0, -15, -3]] % Eigenvalues: 2, 12, -13 U*diag([[20, 120, -130]])*inv( U )/10 = [[2, -4.5, 7.2], [0, -4, 9.6], [0, 15, 3]] % Eigenvalues: 1, -11, 14 U*diag([[10, -110, 140]])*inv( U )/10 = [[1, 3, -7.2], [0, 5, -9.6], [0, -15, -2]] % Eigenvalues: 1, 11, -14 U*diag([[10, 110, -140]])*inv( U )/10 = [[1, -4.5, 7.2], [0, -5, 9.6], [0, 15, 2]] % Eigenvalues: 2, -14, 11 U*diag([[20, -140, 110]])*inv( U )/10 = [[2, 0, -7.2], [0, 2, -9.6], [0, -15, -5]] % Eigenvalues: 2, 14, -11 U*diag([[20, 140, -110]])*inv( U )/10 = [[2, -3, 7.2], [0, -2, 9.6], [0, 15, 5]] % Eigenvalues: 3, -13, 12 U*diag([[30, -130, 120]])*inv( U )/10 = [[3, 0, -7.2], [0, 3, -9.6], [0, -15, -4]] % Eigenvalues: 3, 13, -12 U*diag([[30, 130, -120]])*inv( U )/10 = [[3, -4.5, 7.2], [0, -3, 9.6], [0, 15, 4]] % Eigenvalues: 1, -15, 10 U*diag([[10, -150, 100]])*inv( U )/10 = [[1, 0, -7.2], [0, 1, -9.6], [0, -15, -6]] % Eigenvalues: 3, -11, 14 U*diag([[30, -110, 140]])*inv( U )/10 = [[3, 1.5, -7.2], [0, 5, -9.6], [0, -15, -2]] % Eigenvalues: 1, 15, -10 U*diag([[10, 150, -100]])*inv( U )/10 = [[1, -1.5, 7.2], [0, -1, 9.6], [0, 15, 6]] % Eigenvalues: 2, -10, 15 U*diag([[20, -100, 150]])*inv( U )/10 = [[2, 3, -7.2], [0, 6, -9.6], [0, -15, -1]] % Eigenvalues: 4, -12, 13 U*diag([[40, -120, 130]])*inv( U )/10 = [[4, 0, -7.2], [0, 4, -9.6], [0, -15, -3]] % Eigenvalues: 4, -14, 11 U*diag([[40, -140, 110]])*inv( U )/10 = [[4, -1.5, -7.2], [0, 2, -9.6], [0, -15, -5]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 2 digits % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, -3, 2 U*diag([[100, -300, 200]])*inv( U )/100 = [[1, -.6, -1.44], [0, .2, -1.92], [0, -3, -1.2]] % Eigenvalues: 1, -2, 3 U*diag([[100, -200, 300]])*inv( U )/100 = [[1, .15, -1.44], [0, 1.2, -1.92], [0, -3, -.2]] % Eigenvalues: 2, -2, 3 U*diag([[200, -200, 300]])*inv( U )/100 = [[2, -.6, -1.44], [0, 1.2, -1.92], [0, -3, -.2]] % Eigenvalues: 1, -1, 4 U*diag([[100, -100, 400]])*inv( U )/100 = [[1, .9, -1.44], [0, 2.2, -1.92], [0, -3, .8]] % Eigenvalues: 2, -4, 1 U*diag([[200, -400, 100]])*inv( U )/100 = [[2, -2.1, -1.44], [0, -.8, -1.92], [0, -3, -2.2]] % Eigenvalues: 2, -1, 4 U*diag([[200, -100, 400]])*inv( U )/100 = [[2, .15, -1.44], [0, 2.2, -1.92], [0, -3, .8]] % Eigenvalues: 3, -3, 2 U*diag([[300, -300, 200]])*inv( U )/100 = [[3, -2.1, -1.44], [0, .2, -1.92], [0, -3, -1.2]] % Eigenvalues: 3, -4, 1 U*diag([[300, -400, 100]])*inv( U )/100 = [[3, -2.85, -1.44], [0, -.8, -1.92], [0, -3, -2.2]] % Eigenvalues: 3, -1, 4 U*diag([[300, -100, 400]])*inv( U )/100 = [[3, -.6, -1.44], [0, 2.2, -1.92], [0, -3, .8]] % Eigenvalues: 1, -6, -1 U*diag([[100, -600, -100]])*inv( U )/100 = [[1, -2.85, -1.44], [0, -2.8, -1.92], [0, -3, -4.2]] % Eigenvalues: 2, -6, -1 U*diag([[200, -600, -100]])*inv( U )/100 = [[2, -3.6, -1.44], [0, -2.8, -1.92], [0, -3, -4.2]] % Eigenvalues: 3, -6, -1 U*diag([[300, -600, -100]])*inv( U )/100 = [[3, -4.35, -1.44], [0, -2.8, -1.92], [0, -3, -4.2]] % Eigenvalues: 1, -5, 5 U*diag([[100, -500, 500]])*inv( U )/100 = [[1, .3, -2.88], [0, 1.4, -3.84], [0, -6, -1.4]] % Eigenvalues: 1, -6, 4 U*diag([[100, -600, 400]])*inv( U )/100 = [[1, -.45, -2.88], [0, .4, -3.84], [0, -6, -2.4]] % Eigenvalues: 1, -4, 6 U*diag([[100, -400, 600]])*inv( U )/100 = [[1, 1.05, -2.88], [0, 2.4, -3.84], [0, -6, -.4]] % Eigenvalues: 2, -5, 5 U*diag([[200, -500, 500]])*inv( U )/100 = [[2, -.45, -2.88], [0, 1.4, -3.84], [0, -6, -1.4]]