U = [[.1, .5, .5, .7], [.5, -.1, -.7, .5], [.5, .7, -.1, -.5], [.7, -.5, .5, -.1]] %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Integer matrices % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 45, 25, -25, -5 U*diag([45, 25, -25, -5])*inv( U ) = [[-2, 8, 14, -9], [8, -2, 9, 26], [14, 9, 22, 8], [-9, 26, 8, 22]] % Eigenvalues: 46, 26, -24, -4 U*diag([46, 26, -24, -4])*inv( U ) = [[-1, 8, 14, -9], [8, -1, 9, 26], [14, 9, 23, 8], [-9, 26, 8, 23]] % Eigenvalues: 47, 27, -23, -3 U*diag([47, 27, -23, -3])*inv( U ) = [[0, 8, 14, -9], [8, 0, 9, 26], [14, 9, 24, 8], [-9, 26, 8, 24]] % Eigenvalues: 48, -22, 28, -2 U*diag([48, -22, 28, -2])*inv( U ) = [[1, -7, -6, 16], [-7, 25, 16, 6], [-6, 16, 1, 23], [16, 6, 23, 25]] % Eigenvalues: 49, 29, -21, -1 U*diag([49, 29, -21, -1])*inv( U ) = [[2, 8, 14, -9], [8, 2, 9, 26], [14, 9, 26, 8], [-9, 26, 8, 26]] % Eigenvalues: 49, -31, 19, -1 U*diag([49, -31, 19, -1])*inv( U ) = [[-3, -3, -9, 16], [-3, 21, 16, 9], [-9, 16, -3, 27], [16, 9, 27, 21]] % Eigenvalues: 51, -29, 21, 1 U*diag([51, -29, 21, 1])*inv( U ) = [[-1, -3, -9, 16], [-3, 23, 16, 9], [-9, 16, -1, 27], [16, 9, 27, 23]] % Eigenvalues: 52, -28, 22, 2 U*diag([52, -28, 22, 2])*inv( U ) = [[0, -3, -9, 16], [-3, 24, 16, 9], [-9, 16, 0, 27], [16, 9, 27, 24]] % Eigenvalues: 53, -27, 23, 3 U*diag([53, -27, 23, 3])*inv( U ) = [[1, -3, -9, 16], [-3, 25, 16, 9], [-9, 16, 1, 27], [16, 9, 27, 25]] % Eigenvalues: 49, -41, 9, -1 U*diag([49, -41, 9, -1])*inv( U ) = [[-8, 1, -12, 16], [1, 16, 16, 12], [-12, 16, -8, 31], [16, 12, 31, 16]] % Eigenvalues: 54, -26, 24, 4 U*diag([54, -26, 24, 4])*inv( U ) = [[2, -3, -9, 16], [-3, 26, 16, 9], [-9, 16, 2, 27], [16, 9, 27, 26]] % Eigenvalues: 51, -39, 11, 1 U*diag([51, -39, 11, 1])*inv( U ) = [[-6, 1, -12, 16], [1, 18, 16, 12], [-12, 16, -6, 31], [16, 12, 31, 18]] % Eigenvalues: 52, -38, 12, 2 U*diag([52, -38, 12, 2])*inv( U ) = [[-5, 1, -12, 16], [1, 19, 16, 12], [-12, 16, -5, 31], [16, 12, 31, 19]] % Eigenvalues: 53, -37, 13, 3 U*diag([53, -37, 13, 3])*inv( U ) = [[-4, 1, -12, 16], [1, 20, 16, 12], [-12, 16, -4, 31], [16, 12, 31, 20]] % Eigenvalues: 54, -36, 14, 4 U*diag([54, -36, 14, 4])*inv( U ) = [[-3, 1, -12, 16], [1, 21, 16, 12], [-12, 16, -3, 31], [16, 12, 31, 21]] % Eigenvalues: 55, -35, 15, 5 U*diag([55, -35, 15, 5])*inv( U ) = [[-2, 1, -12, 16], [1, 22, 16, 12], [-12, 16, -2, 31], [16, 12, 31, 22]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 1 digit % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 3, -1, 4, -2 U*diag([[30, -10, 40, -20]])*inv( U )/10 = [[-.2, -1.9, .3, 1.6], [-1.9, 2.2, 1.6, -.3], [.3, 1.6, -.2, 1.1], [1.6, -.3, 1.1, 2.2]] % Eigenvalues: 6, -3, 2, 1 U*diag([[60, -30, 20, 10]])*inv( U )/10 = [[.3, .1, -1.2, 1.6], [.1, 2.7, 1.6, 1.2], [-1.2, 1.6, .3, 3.1], [1.6, 1.2, 3.1, 2.7]] % Eigenvalues: 7, -2, 3, 2 U*diag([[70, -20, 30, 20]])*inv( U )/10 = [[1.3, .1, -1.2, 1.6], [.1, 3.7, 1.6, 1.2], [-1.2, 1.6, 1.3, 3.1], [1.6, 1.2, 3.1, 3.7]] % Eigenvalues: 7, -4, 1, 2 U*diag([[70, -40, 10, 20]])*inv( U )/10 = [[.3, .9, -1.8, 1.6], [.9, 2.7, 1.6, 1.8], [-1.8, 1.6, .3, 3.9], [1.6, 1.8, 3.9, 2.7]] % Eigenvalues: 4, 8, 3, -1 U*diag([[40, 80, 30, -10]])*inv( U )/10 = [[2.3, -1.6, 3.2, -.9], [-1.6, 2.3, .9, .8], [3.2, .9, 4.7, -1.6], [-.9, .8, -1.6, 4.7]] % Eigenvalues: 8, -5, 5, -2 U*diag([[80, -50, 50, -20]])*inv( U )/10 = [[-.9, -1.8, -.9, 3.2], [-1.8, 3.9, 3.2, .9], [-.9, 3.2, -.9, 4.2], [3.2, .9, 4.2, 3.9]] % Eigenvalues: 9, -5, 5, -1 U*diag([[90, -50, 50, -10]])*inv( U )/10 = [[-.4, -1.4, -1.2, 3.2], [-1.4, 4.4, 3.2, 1.2], [-1.2, 3.2, -.4, 4.6], [3.2, 1.2, 4.6, 4.4]] % Eigenvalues: 9, -4, 6, -1 U*diag([[90, -40, 60, -10]])*inv( U )/10 = [[.1, -1.8, -.9, 3.2], [-1.8, 4.9, 3.2, .9], [-.9, 3.2, .1, 4.2], [3.2, .9, 4.2, 4.9]] % Eigenvalues: 4, 10, 5, -1 U*diag([[40, 100, 50, -10]])*inv( U )/10 = [[3.3, -2.4, 3.8, -.9], [-2.4, 3.3, .9, .2], [3.8, .9, 5.7, -2.4], [-.9, .2, -2.4, 5.7]] % Eigenvalues: 11, -5, 5, 1 U*diag([[110, -50, 50, 10]])*inv( U )/10 = [[.6, -.6, -1.8, 3.2], [-.6, 5.4, 3.2, 1.8], [-1.8, 3.2, .6, 5.4], [3.2, 1.8, 5.4, 5.4]] % Eigenvalues: 11, -6, 4, 1 U*diag([[110, -60, 40, 10]])*inv( U )/10 = [[.1, -.2, -2.1, 3.2], [-.2, 4.9, 3.2, 2.1], [-2.1, 3.2, .1, 5.8], [3.2, 2.1, 5.8, 4.9]] % Eigenvalues: 4, 11, 6, -1 U*diag([[40, 110, 60, -10]])*inv( U )/10 = [[3.8, -2.8, 4.1, -.9], [-2.8, 3.8, .9, -.1], [4.1, .9, 6.2, -2.8], [-.9, -.1, -2.8, 6.2]] % Eigenvalues: 13, -3, 2, -2 U*diag([[130, -30, 20, -20]])*inv( U )/10 = [[-1.1, -.6, .2, 2.3], [-.6, 3.7, 4.1, 3.8], [.2, 4.1, 1.3, 5.4], [2.3, 3.8, 5.4, 6.1]] % Eigenvalues: 6, -12, 3, 1 U*diag([[60, -120, 30, 10]])*inv( U )/10 = [[-1.7, .2, -4.4, 4.1], [.2, 3.1, 2.3, .4], [-4.4, 2.3, -4.1, 6.2], [4.1, .4, 6.2, .7]] % Eigenvalues: 13, -4, 1, -2 U*diag([[130, -40, 10, -20]])*inv( U )/10 = [[-1.6, -.2, -.1, 2.3], [-.2, 3.2, 4.1, 4.1], [-.1, 4.1, .8, 5.8], [2.3, 4.1, 5.8, 5.6]] % Eigenvalues: 13, -1, 4, -2 U*diag([[130, -10, 40, -20]])*inv( U )/10 = [[-.1, -1.4, .8, 2.3], [-1.4, 4.7, 4.1, 3.2], [.8, 4.1, 2.3, 4.6], [2.3, 3.2, 4.6, 7.1]] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrices with at most 2 digits % % beyond the decimal point % % with integer eigenvalues % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Eigenvalues: 1, -1, -2, 2 U*diag([[100, -100, -200, 200]])*inv( U )/100 = [[.24, 1.5, -.9, -.32], [1.5, -.24, -.32, .9], [-.9, -.32, .24, .9], [-.32, .9, .9, -.24]] % Eigenvalues: 3, 1, -1, 2 U*diag([[300, 100, -100, 200]])*inv( U )/100 = [[1.01, 1.15, -.15, -.43], [1.15, .77, .11, 1.35], [-.15, .11, 1.73, .85], [-.43, 1.35, .85, 1.49]] % Eigenvalues: 3, -2, -1, 1 U*diag([[300, -200, -100, 100]])*inv( U )/100 = [[-.23, .95, -.85, .39], [.95, .49, .57, 1.25], [-.85, .57, .01, 1.85], [.39, 1.25, 1.85, .73]] % Eigenvalues: 2, 1, -3, -1 U*diag([[200, 100, -300, -100]])*inv( U )/100 = [[-.97, .75, .95, -.79], [.75, -1.21, .47, 1.85], [.95, .47, .71, .45], [-.79, 1.85, .45, .47]] % Eigenvalues: 3, -2, -1, 2 U*diag([[300, -200, -100, 200]])*inv( U )/100 = [[.26, 1.3, -1.2, .32], [1.3, .74, .32, 1.2], [-1.2, .32, .26, 1.9], [.32, 1.2, 1.9, .74]] % Eigenvalues: 2, 1, -3, -2 U*diag([[200, 100, -300, -200]])*inv( U )/100 = [[-1.46, .4, 1.3, -.72], [.4, -1.46, .72, 1.9], [1.3, .72, .46, .4], [-.72, 1.9, .4, .46]] % Eigenvalues: 3, 1, -3, -1 U*diag([[300, 100, -300, -100]])*inv( U )/100 = [[-.96, .8, 1, -.72], [.8, -.96, .72, 2.2], [1, .72, .96, .8], [-.72, 2.2, .8, .96]] % Eigenvalues: 4, 1, -1, 2 U*diag([[400, 100, -100, 200]])*inv( U )/100 = [[1.02, 1.2, -.1, -.36], [1.2, 1.02, .36, 1.7], [-.1, .36, 1.98, 1.2], [-.36, 1.7, 1.2, 1.98]] % Eigenvalues: 4, -1, -2, 1 U*diag([[400, -100, -200, 100]])*inv( U )/100 = [[-.22, 1.3, -.4, -.04], [1.3, .26, .68, 2], [-.4, .68, .74, 1.9], [-.04, 2, 1.9, 1.22]] % Eigenvalues: 3, 1, -3, 2 U*diag([[300, 100, -300, 200]])*inv( U )/100 = [[.51, 1.85, -.05, -.93], [1.85, -.21, -.03, 2.05], [-.05, -.03, 1.71, .95], [-.93, 2.05, .95, .99]] % Eigenvalues: 3, 1, -3, -2 U*diag([[300, 100, -300, -200]])*inv( U )/100 = [[-1.45, .45, 1.35, -.65], [.45, -1.21, .97, 2.25], [1.35, .97, .71, .75], [-.65, 2.25, .75, .95]] % Eigenvalues: 4, 1, -2, 2 U*diag([[400, 100, -200, 200]])*inv( U )/100 = [[.77, 1.55, -.05, -.61], [1.55, .53, .29, 2.05], [-.05, .29, 1.97, 1.25], [-.61, 2.05, 1.25, 1.73]] % Eigenvalues: 4, -1, -2, 2 U*diag([[400, -100, -200, 200]])*inv( U )/100 = [[.27, 1.65, -.75, -.11], [1.65, .51, .43, 1.95], [-.75, .43, .99, 1.95], [-.11, 1.95, 1.95, 1.23]] % Eigenvalues: 4, 1, -3, -1 U*diag([[400, 100, -300, -100]])*inv( U )/100 = [[-.95, .85, 1.05, -.65], [.85, -.71, .97, 2.55], [1.05, .97, 1.21, 1.15], [-.65, 2.55, 1.15, 1.45]] % Eigenvalues: 3, 1, -4, -1 U*diag([[300, 100, -400, -100]])*inv( U )/100 = [[-1.21, 1.15, 1.05, -.97], [1.15, -1.45, .65, 2.55], [1.05, .65, .95, .85], [-.97, 2.55, .85, .71]] % Eigenvalues: 4, 1, -3, 2 U*diag([[400, 100, -300, 200]])*inv( U )/100 = [[.52, 1.9, 0, -.86], [1.9, .04, .22, 2.4], [0, .22, 1.96, 1.3], [-.86, 2.4, 1.3, 1.48]]