U = [[1, .6, .6], [0, .48, .64], [0, .64, -.48]] inv( U ) = [[1, -1.05, -.15], [0, .75, 1], [0, 1, -.75]] %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Integer matrices % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 7, -53, 47 U*diag([[7, -53, 47]])*inv( U ) = [[7, -3, -54], [0, 11, -48], [0, -48, -17]] % Eigenvalues: 6, -54, 46 U*diag([[6, -54, 46]])*inv( U ) = [[6, -3, -54], [0, 10, -48], [0, -48, -18]] % Eigenvalues: 8, -52, 48 U*diag([[8, -52, 48]])*inv( U ) = [[8, -3, -54], [0, 12, -48], [0, -48, -16]] % Eigenvalues: 5, -55, 45 U*diag([[5, -55, 45]])*inv( U ) = [[5, -3, -54], [0, 9, -48], [0, -48, -19]] % Eigenvalues: 9, -51, 49 U*diag([[9, -51, 49]])*inv( U ) = [[9, -3, -54], [0, 13, -48], [0, -48, -15]] % Eigenvalues: 4, -56, 44 U*diag([[4, -56, 44]])*inv( U ) = [[4, -3, -54], [0, 8, -48], [0, -48, -20]] % Eigenvalues: 3, -57, 43 U*diag([[3, -57, 43]])*inv( U ) = [[3, -3, -54], [0, 7, -48], [0, -48, -21]] % Eigenvalues: 2, -58, 42 U*diag([[2, -58, 42]])*inv( U ) = [[2, -3, -54], [0, 6, -48], [0, -48, -22]] % Eigenvalues: 1, -59, 41 U*diag([[1, -59, 41]])*inv( U ) = [[1, -3, -54], [0, 5, -48], [0, -48, -23]] % Eigenvalues: 1, 61, -39 U*diag([[1, 61, -39]])*inv( U ) = [[1, 3, 54], [0, -3, 48], [0, 48, 25]] % Eigenvalues: 2, 62, -38 U*diag([[2, 62, -38]])*inv( U ) = [[2, 3, 54], [0, -2, 48], [0, 48, 26]] % Eigenvalues: 3, 63, -37 U*diag([[3, 63, -37]])*inv( U ) = [[3, 3, 54], [0, -1, 48], [0, 48, 27]] % Eigenvalues: 7, -113, 87 U*diag([[7, -113, 87]])*inv( U ) = [[7, -6, -108], [0, 15, -96], [0, -96, -41]] % Eigenvalues: 6, -114, 86 U*diag([[6, -114, 86]])*inv( U ) = [[6, -6, -108], [0, 14, -96], [0, -96, -42]] % Eigenvalues: 5, -115, 85 U*diag([[5, -115, 85]])*inv( U ) = [[5, -6, -108], [0, 13, -96], [0, -96, -43]] % Eigenvalues: 4, -116, 84 U*diag([[4, -116, 84]])*inv( U ) = [[4, -6, -108], [0, 12, -96], [0, -96, -44]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 1 digit % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, -5, 5 U*diag([[10, -50, 50]])*inv( U )/10 = [[1, -.3, -5.4], [0, 1.4, -4.8], [0, -4.8, -1.4]] % Eigenvalues: 2, -4, 6 U*diag([[20, -40, 60]])*inv( U )/10 = [[2, -.3, -5.4], [0, 2.4, -4.8], [0, -4.8, -.4]] % Eigenvalues: 1, 7, -3 U*diag([[10, 70, -30]])*inv( U )/10 = [[1, .3, 5.4], [0, .6, 4.8], [0, 4.8, 3.4]] % Eigenvalues: 3, 7, -3 U*diag([[30, 70, -30]])*inv( U )/10 = [[3, -1.8, 5.1], [0, .6, 4.8], [0, 4.8, 3.4]] % Eigenvalues: 2, 8, -2 U*diag([[20, 80, -20]])*inv( U )/10 = [[2, .3, 5.4], [0, 1.6, 4.8], [0, 4.8, 4.4]] % Eigenvalues: 4, 8, -2 U*diag([[40, 80, -20]])*inv( U )/10 = [[4, -1.8, 5.1], [0, 1.6, 4.8], [0, 4.8, 4.4]] % Eigenvalues: 3, 9, -1 U*diag([[30, 90, -10]])*inv( U )/10 = [[3, .3, 5.4], [0, 2.6, 4.8], [0, 4.8, 5.4]] % Eigenvalues: 5, 11, 1 U*diag([[50, 110, 10]])*inv( U )/10 = [[5, .3, 5.4], [0, 4.6, 4.8], [0, 4.8, 7.4]] % Eigenvalues: 6, 12, 2 U*diag([[60, 120, 20]])*inv( U )/10 = [[6, .3, 5.4], [0, 5.6, 4.8], [0, 4.8, 8.4]] % Eigenvalues: 1, -11, 9 U*diag([[10, -110, 90]])*inv( U )/10 = [[1, -.6, -10.8], [0, 1.8, -9.6], [0, -9.6, -3.8]] % Eigenvalues: 2, -10, 10 U*diag([[20, -100, 100]])*inv( U )/10 = [[2, -.6, -10.8], [0, 2.8, -9.6], [0, -9.6, -2.8]] % Eigenvalues: 3, -9, 11 U*diag([[30, -90, 110]])*inv( U )/10 = [[3, -.6, -10.8], [0, 3.8, -9.6], [0, -9.6, -1.8]] % Eigenvalues: 4, -8, 12 U*diag([[40, -80, 120]])*inv( U )/10 = [[4, -.6, -10.8], [0, 4.8, -9.6], [0, -9.6, -.8]] % Eigenvalues: 7, 17, -3 U*diag([[70, 170, -30]])*inv( U )/10 = [[7, -1.5, 10.5], [0, 4.2, 9.6], [0, 9.6, 9.8]] % Eigenvalues: 8, 18, -2 U*diag([[80, 180, -20]])*inv( U )/10 = [[8, -1.5, 10.5], [0, 5.2, 9.6], [0, 9.6, 10.8]] % Eigenvalues: 9, 19, -1 U*diag([[90, 190, -10]])*inv( U )/10 = [[9, -1.5, 10.5], [0, 6.2, 9.6], [0, 9.6, 11.8]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 2 digits % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, 2, -1 U*diag([[100, 200, -100]])*inv( U )/100 = [[1, -.75, 1.5], [0, .08, 1.44], [0, 1.44, .92]] % Eigenvalues: 1, 2, -2 U*diag([[100, 200, -200]])*inv( U )/100 = [[1, -1.35, 1.95], [0, -.56, 1.92], [0, 1.92, .56]] % Eigenvalues: 1, 3, -1 U*diag([[100, 300, -100]])*inv( U )/100 = [[1, -.3, 2.1], [0, .44, 1.92], [0, 1.92, 1.56]] % Eigenvalues: 1, 2, -3 U*diag([[100, 200, -300]])*inv( U )/100 = [[1, -1.95, 2.4], [0, -1.2, 2.4], [0, 2.4, .2]] % Eigenvalues: 1, 3, -2 U*diag([[100, 300, -200]])*inv( U )/100 = [[1, -.9, 2.55], [0, -.2, 2.4], [0, 2.4, 1.2]] % Eigenvalues: 2, 3, 1 U*diag([[200, 300, 100]])*inv( U )/100 = [[2, -.15, 1.05], [0, 1.72, .96], [0, .96, 2.28]] % Eigenvalues: 1, 4, -1 U*diag([[100, 400, -100]])*inv( U )/100 = [[1, .15, 2.7], [0, .8, 2.4], [0, 2.4, 2.2]] % Eigenvalues: 1, 3, -3 U*diag([[100, 300, -300]])*inv( U )/100 = [[1, -1.5, 3], [0, -.84, 2.88], [0, 2.88, .84]] % Eigenvalues: 2, 4, -1 U*diag([[200, 400, -100]])*inv( U )/100 = [[2, -.9, 2.55], [0, .8, 2.4], [0, 2.4, 2.2]] % Eigenvalues: 1, 4, -2 U*diag([[100, 400, -200]])*inv( U )/100 = [[1, -.45, 3.15], [0, .16, 2.88], [0, 2.88, 1.84]] % Eigenvalues: 2, 4, -2 U*diag([[200, 400, -200]])*inv( U )/100 = [[2, -1.5, 3], [0, .16, 2.88], [0, 2.88, 1.84]] % Eigenvalues: 3, 1, 4 U*diag([[300, 100, 400]])*inv( U )/100 = [[3, -.3, -1.65], [0, 2.92, -1.44], [0, -1.44, 2.08]] % Eigenvalues: 2, 4, -3 U*diag([[200, 400, -300]])*inv( U )/100 = [[2, -2.1, 3.45], [0, -.48, 3.36], [0, 3.36, 1.48]] % Eigenvalues: 2, 5, -1 U*diag([[200, 500, -100]])*inv( U )/100 = [[2, -.45, 3.15], [0, 1.16, 2.88], [0, 2.88, 2.84]] % Eigenvalues: 2, 5, -2 U*diag([[200, 500, -200]])*inv( U )/100 = [[2, -1.05, 3.6], [0, .52, 3.36], [0, 3.36, 2.48]] % Eigenvalues: 2, 5, -3 U*diag([[200, 500, -300]])*inv( U )/100 = [[2, -1.65, 4.05], [0, -.12, 3.84], [0, 3.84, 2.12]]