U = [[.6, .28, .36], [.8, -.96, .48], [0, 0, .8]] inv( U ) = [[1.2, .35, -.75], [1, -.75, 0], [0, 0, 1.25]] %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Integer matrices % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 47, -53, 7 U*diag([[47, -53, 7]])*inv( U ) = [[19, 21, -18], [96, -25, -24], [0, 0, 7]] % Eigenvalues: 53, -47, -7 U*diag([[53, -47, -7]])*inv( U ) = [[25, 21, -27], [96, -19, -36], [0, 0, -7]] % Eigenvalues: 46, -54, 6 U*diag([[46, -54, 6]])*inv( U ) = [[18, 21, -18], [96, -26, -24], [0, 0, 6]] % Eigenvalues: 54, -46, -6 U*diag([[54, -46, -6]])*inv( U ) = [[26, 21, -27], [96, -18, -36], [0, 0, -6]] % Eigenvalues: 48, -52, 8 U*diag([[48, -52, 8]])*inv( U ) = [[20, 21, -18], [96, -24, -24], [0, 0, 8]] % Eigenvalues: 52, -48, -8 U*diag([[52, -48, -8]])*inv( U ) = [[24, 21, -27], [96, -20, -36], [0, 0, -8]] % Eigenvalues: 45, -55, 5 U*diag([[45, -55, 5]])*inv( U ) = [[17, 21, -18], [96, -27, -24], [0, 0, 5]] % Eigenvalues: 55, -45, -5 U*diag([[55, -45, -5]])*inv( U ) = [[27, 21, -27], [96, -17, -36], [0, 0, -5]] % Eigenvalues: 49, -51, 9 U*diag([[49, -51, 9]])*inv( U ) = [[21, 21, -18], [96, -23, -24], [0, 0, 9]] % Eigenvalues: 51, -49, -9 U*diag([[51, -49, -9]])*inv( U ) = [[23, 21, -27], [96, -21, -36], [0, 0, -9]] % Eigenvalues: 44, -56, 4 U*diag([[44, -56, 4]])*inv( U ) = [[16, 21, -18], [96, -28, -24], [0, 0, 4]] % Eigenvalues: 50, -50, -10 U*diag([[50, -50, -10]])*inv( U ) = [[22, 21, -27], [96, -22, -36], [0, 0, -10]] % Eigenvalues: 50, -50, 10 U*diag([[50, -50, 10]])*inv( U ) = [[22, 21, -18], [96, -22, -24], [0, 0, 10]] % Eigenvalues: 49, -51, -11 U*diag([[49, -51, -11]])*inv( U ) = [[21, 21, -27], [96, -23, -36], [0, 0, -11]] % Eigenvalues: 48, -52, -12 U*diag([[48, -52, -12]])*inv( U ) = [[20, 21, -27], [96, -24, -36], [0, 0, -12]] % Eigenvalues: 47, -53, -13 U*diag([[47, -53, -13]])*inv( U ) = [[19, 21, -27], [96, -25, -36], [0, 0, -13]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 1 digit % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 5, -5, -1 U*diag([[50, -50, -10]])*inv( U )/10 = [[2.2, 2.1, -2.7], [9.6, -2.2, -3.6], [0, 0, -1]] % Eigenvalues: 5, -5, 1 U*diag([[50, -50, 10]])*inv( U )/10 = [[2.2, 2.1, -1.8], [9.6, -2.2, -2.4], [0, 0, 1]] % Eigenvalues: 6, -4, -2 U*diag([[60, -40, -20]])*inv( U )/10 = [[3.2, 2.1, -3.6], [9.6, -1.2, -4.8], [0, 0, -2]] % Eigenvalues: 4, -6, -2 U*diag([[40, -60, -20]])*inv( U )/10 = [[1.2, 2.1, -2.7], [9.6, -3.2, -3.6], [0, 0, -2]] % Eigenvalues: 6, -4, 2 U*diag([[60, -40, 20]])*inv( U )/10 = [[3.2, 2.1, -1.8], [9.6, -1.2, -2.4], [0, 0, 2]] % Eigenvalues: 4, -6, 2 U*diag([[40, -60, 20]])*inv( U )/10 = [[1.2, 2.1, -.9], [9.6, -3.2, -1.2], [0, 0, 2]] % Eigenvalues: 5, -5, -3 U*diag([[50, -50, -30]])*inv( U )/10 = [[2.2, 2.1, -3.6], [9.6, -2.2, -4.8], [0, 0, -3]] % Eigenvalues: 7, -3, -1 U*diag([[70, -30, -10]])*inv( U )/10 = [[4.2, 2.1, -3.6], [9.6, -.2, -4.8], [0, 0, -1]] % Eigenvalues: 7, -3, 1 U*diag([[70, -30, 10]])*inv( U )/10 = [[4.2, 2.1, -2.7], [9.6, -.2, -3.6], [0, 0, 1]] % Eigenvalues: 3, -7, -1 U*diag([[30, -70, -10]])*inv( U )/10 = [[.2, 2.1, -1.8], [9.6, -4.2, -2.4], [0, 0, -1]] % Eigenvalues: 3, -7, 1 U*diag([[30, -70, 10]])*inv( U )/10 = [[.2, 2.1, -.9], [9.6, -4.2, -1.2], [0, 0, 1]] % Eigenvalues: 3, -7, -3 U*diag([[30, -70, -30]])*inv( U )/10 = [[.2, 2.1, -2.7], [9.6, -4.2, -3.6], [0, 0, -3]] % Eigenvalues: 4, -6, -4 U*diag([[40, -60, -40]])*inv( U )/10 = [[1.2, 2.1, -3.6], [9.6, -3.2, -4.8], [0, 0, -4]] % Eigenvalues: 8, -2, 2 U*diag([[80, -20, 20]])*inv( U )/10 = [[5.2, 2.1, -2.7], [9.6, .8, -3.6], [0, 0, 2]] % Eigenvalues: 2, -8, -2 U*diag([[20, -80, -20]])*inv( U )/10 = [[-.8, 2.1, -1.8], [9.6, -5.2, -2.4], [0, 0, -2]] % Eigenvalues: 3, -7, -5 U*diag([[30, -70, -50]])*inv( U )/10 = [[.2, 2.1, -3.6], [9.6, -4.2, -4.8], [0, 0, -5]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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 = [[.16, .63, -.9], [2.88, -1.16, -1.2], [0, 0, -1]] % Eigenvalues: 1, -1, 2 U*diag([[100, -100, 200]])*inv( U )/100 = [[.44, .42, .45], [1.92, -.44, .6], [0, 0, 2]] % Eigenvalues: 1, -2, 2 U*diag([[100, -200, 200]])*inv( U )/100 = [[.16, .63, .45], [2.88, -1.16, .6], [0, 0, 2]] % Eigenvalues: 2, -2, -1 U*diag([[200, -200, -100]])*inv( U )/100 = [[.88, .84, -1.35], [3.84, -.88, -1.8], [0, 0, -1]] % Eigenvalues: 1, -3, -1 U*diag([[100, -300, -100]])*inv( U )/100 = [[-.12, .84, -.9], [3.84, -1.88, -1.2], [0, 0, -1]] % Eigenvalues: 1, -1, 3 U*diag([[100, -100, 300]])*inv( U )/100 = [[.44, .42, .9], [1.92, -.44, 1.2], [0, 0, 3]] % Eigenvalues: 1, -3, -2 U*diag([[100, -300, -200]])*inv( U )/100 = [[-.12, .84, -1.35], [3.84, -1.88, -1.8], [0, 0, -2]] % Eigenvalues: 1, -3, 2 U*diag([[100, -300, 200]])*inv( U )/100 = [[-.12, .84, .45], [3.84, -1.88, .6], [0, 0, 2]] % Eigenvalues: 1, -2, 3 U*diag([[100, -200, 300]])*inv( U )/100 = [[.16, .63, .9], [2.88, -1.16, 1.2], [0, 0, 3]] % Eigenvalues: 1, 2, 3 U*diag([[100, 200, 300]])*inv( U )/100 = [[1.28, -.21, .9], [-.96, 1.72, 1.2], [0, 0, 3]] % Eigenvalues: 2, -3, -1 U*diag([[200, -300, -100]])*inv( U )/100 = [[.6, 1.05, -1.35], [4.8, -1.6, -1.8], [0, 0, -1]] % Eigenvalues: 2, -1, 3 U*diag([[200, -100, 300]])*inv( U )/100 = [[1.16, .63, .45], [2.88, -.16, .6], [0, 0, 3]] % Eigenvalues: 2, -3, -2 U*diag([[200, -300, -200]])*inv( U )/100 = [[.6, 1.05, -1.8], [4.8, -1.6, -2.4], [0, 0, -2]] % Eigenvalues: 2, -2, 3 U*diag([[200, -200, 300]])*inv( U )/100 = [[.88, .84, .45], [3.84, -.88, .6], [0, 0, 3]] % Eigenvalues: 1, -3, 3 U*diag([[100, -300, 300]])*inv( U )/100 = [[-.12, .84, .9], [3.84, -1.88, 1.2], [0, 0, 3]] % Eigenvalues: 2, -3, 3 U*diag([[200, -300, 300]])*inv( U )/100 = [[.6, 1.05, .45], [4.8, -1.6, .6], [0, 0, 3]]