octave:2> A=rand(4) A = 0.378725 0.444359 0.816689 0.314153 0.755537 0.083429 0.051877 0.134796 0.361388 0.944693 0.543669 0.485129 0.975967 0.575803 0.096310 0.504597 octave:3> b=rand(4,1) b = 0.059054 0.202029 0.954583 0.402325 octave:4> A\b ans = 0.60913 2.79538 -0.38594 -3.49702 octave:5> x=A\b x = 0.60913 2.79538 -0.38594 -3.49702 octave:6> A*x ans = 0.059054 0.202029 0.954583 0.402325 octave:7> b b = 0.059054 0.202029 0.954583 0.402325 octave:8> y=b'/A y = 2.1521 -2.2464 -1.5341 1.5325 octave:9> y*A ans = 0.059054 0.202029 0.954583 0.402325 octave:10> b' ans = 0.059054 0.202029 0.954583 0.402325 octave:11> A'\b ans = 2.1521 -2.2464 -1.5341 1.5325 octave:12> (A'\b)' ans = 2.1521 -2.2464 -1.5341 1.5325 octave:13> y y = 2.1521 -2.2464 -1.5341 1.5325 octave:14> A/b error: operator /: nonconformant arguments (op1 is 4x4, op2 is 4x1) octave:14> A/b' ans = 0.91116 0.14787 0.82903 0.41965 octave:15> b'/A ans = 2.1521 -2.2464 -1.5341 1.5325 octave:16> A=rand(4) A = 0.333816 0.289379 0.038941 0.243195 0.093559 0.865393 0.495694 0.441394 0.251444 0.125429 0.809112 0.832125 0.977261 0.657617 0.928700 0.306450 octave:17> [V,D]=eig(A) V = Columns 1 through 3: 0.19774 + 0.00000i 0.19228 + 0.00000i -0.44568 + 0.25594i 0.51515 + 0.00000i 0.08258 + 0.00000i -0.34007 - 0.38876i 0.54940 + 0.00000i 0.48825 + 0.00000i 0.65707 + 0.00000i 0.62744 + 0.00000i -0.84725 + 0.00000i -0.11270 + 0.15698i Column 4: -0.44568 - 0.25594i -0.34007 + 0.38876i 0.65707 - 0.00000i -0.11270 - 0.15698i D = Diagonal Matrix Columns 1 through 3: 1.96755 + 0.00000i 0 0 0 -0.51462 + 0.00000i 0 0 0 0.43092 + 0.22253i 0 0 0 Column 4: 0 0 0 0.43092 - 0.22253i octave:18> octave:18> A*V ans = Columns 1 through 3: 0.38907 + 0.00000i -0.09895 + 0.00000i -0.24900 + 0.01111i 1.01359 + 0.00000i -0.04250 + 0.00000i -0.06003 - 0.24320i 1.08097 + 0.00000i -0.25126 + 0.00000i 0.28314 + 0.14622i 1.23452 + 0.00000i 0.43601 + 0.00000i -0.08350 + 0.04257i Column 4: -0.24900 - 0.01111i -0.06003 + 0.24320i 0.28314 - 0.14622i -0.08350 - 0.04257i octave:19> V*D ans = Columns 1 through 3: 0.38907 + 0.00000i -0.09895 + 0.00000i -0.24900 + 0.01111i 1.01359 + 0.00000i -0.04250 + 0.00000i -0.06003 - 0.24320i 1.08097 + 0.00000i -0.25126 + 0.00000i 0.28314 + 0.14622i 1.23452 + 0.00000i 0.43601 - 0.00000i -0.08350 + 0.04257i Column 4: -0.24900 - 0.01111i -0.06003 + 0.24320i 0.28314 - 0.14622i -0.08350 - 0.04257i octave:20> [V,D]=eig([1,1;0,1]) V = 1.00000 -1.00000 0.00000 0.00000 D = Diagonal Matrix 1 0 0 1 octave:21> [V,D]=eig(A) V = Columns 1 through 3: 0.19774 + 0.00000i 0.19228 + 0.00000i -0.44568 + 0.25594i 0.51515 + 0.00000i 0.08258 + 0.00000i -0.34007 - 0.38876i 0.54940 + 0.00000i 0.48825 + 0.00000i 0.65707 + 0.00000i 0.62744 + 0.00000i -0.84725 + 0.00000i -0.11270 + 0.15698i Column 4: -0.44568 - 0.25594i -0.34007 + 0.38876i 0.65707 - 0.00000i -0.11270 - 0.15698i D = Diagonal Matrix Columns 1 through 3: 1.96755 + 0.00000i 0 0 0 -0.51462 + 0.00000i 0 0 0 0.43092 + 0.22253i 0 0 0 Column 4: 0 0 0 0.43092 - 0.22253i octave:22> norm(V) ans = 1.3295 octave:23> norm(V(:,1)) ans = 1 octave:24> norm(V(:,2)) ans = 1 octave:25> norm(V(:,3)) ans = 1.00000 octave:26> norm(V(:,4)) ans = 1.00000 octave:27> eig([0,1;1,0]) ans = -1 1 octave:28> P=rand(2) P = 0.40538 0.14254 0.12345 0.19615 octave:29> P=rand(4) P = 0.349360 0.153547 0.063192 0.101390 0.376376 0.058941 0.754478 0.908796 0.122424 0.692081 0.978824 0.983139 0.295750 0.529396 0.287781 0.198474 octave:30> sum(P) ans = 1.1439 1.4340 2.0843 2.1918 octave:31> sum(P,2) ans = 0.66749 2.09859 2.77647 1.31140 octave:32> diag(sum(P,2)) ans = Diagonal Matrix 0.66749 0 0 0 0 2.09859 0 0 0 0 2.77647 0 0 0 0 1.31140 octave:33> spdiags(sum(P,2),0,4,4) ans = Compressed Column Sparse (rows = 4, cols = 4, nnz = 4 [25%]) (1, 1) -> 0.66749 (2, 2) -> 2.0986 (3, 3) -> 2.7765 (4, 4) -> 1.3114 octave:34> spdiags(sum(P,2),0,4,4)\P ans = 0.523395 0.230037 0.094672 0.151897 0.179347 0.028086 0.359516 0.433051 0.044093 0.249267 0.352543 0.354097 0.225522 0.403687 0.219445 0.151345 octave:35> sum(ans,2) ans = 1.00000 1.00000 1.00000 1.00000 octave:36> P=randstoc(4) P = 0.108326 0.479616 0.400215 0.011842 0.439471 0.375577 0.138588 0.046364 0.012442 0.297102 0.197041 0.493414 0.128388 0.305225 0.422169 0.144218 octave:37> sum(P,2) ans = 1 1 1 1 octave:38> v=invariantPotenze(P,1e-6,100) v = 0.20807 0.36512 0.25635 0.17046 octave:39> v*P ans = 0.20807 0.36512 0.25635 0.17046 octave:40> fomat long e error: 'fomat' undefined near line 1 column 1 octave:40> format long e octave:41> v v = Columns 1 through 3: 2.08072033847156e-01 3.65115123083029e-01 2.56349944238025e-01 Column 4: 1.70462898831790e-01 octave:42> v*P ans = Columns 1 through 3: 2.08072060667042e-01 3.65115145919567e-01 2.56349896142645e-01 Column 4: 1.70462897270746e-01 octave:43> [V,D]=eig(P') V = Column 1: -3.99418466784413e-01 + 0.00000000000000e+00i -7.00880862441566e-01 + 0.00000000000000e+00i -4.92093340269944e-01 + 0.00000000000000e+00i -3.27223241101125e-01 + 0.00000000000000e+00i Column 2: 2.11690442538158e-01 + 3.29361627516217e-01i -4.34712145592423e-02 - 2.06571454989098e-01i 5.37549569786568e-01 - 1.22790172527118e-01i -7.05768797765483e-01 + 0.00000000000000e+00i Column 3: 2.11690442538158e-01 - 3.29361627516217e-01i -4.34712145592423e-02 + 2.06571454989098e-01i 5.37549569786568e-01 + 1.22790172527118e-01i -7.05768797765483e-01 - 0.00000000000000e+00i Column 4: -5.23358463054956e-01 + 0.00000000000000e+00i -4.25526695575711e-01 + 0.00000000000000e+00i 2.56691082918179e-01 + 0.00000000000000e+00i 6.92194075712487e-01 + 0.00000000000000e+00i D = Diagonal Matrix Column 1: 9.99999999999999e-01 + 0.00000000000000e+00i 0 0 0 Column 2: 0 -2.32287564388187e-01 + 9.38883338238302e-02i 0 0 Column 3: 0 0 -2.32287564388187e-01 - 9.38883338238302e-02i 0 Column 4: 0 0 0 2.89738284189780e-01 + 0.00000000000000e+00i octave:44> format short e octave:45> [V,D]=eig(P') V = Columns 1 and 2: -3.9942e-01 + 0.0000e+00i 2.1169e-01 + 3.2936e-01i -7.0088e-01 + 0.0000e+00i -4.3471e-02 - 2.0657e-01i -4.9209e-01 + 0.0000e+00i 5.3755e-01 - 1.2279e-01i -3.2722e-01 + 0.0000e+00i -7.0577e-01 + 0.0000e+00i Columns 3 and 4: 2.1169e-01 - 3.2936e-01i -5.2336e-01 + 0.0000e+00i -4.3471e-02 + 2.0657e-01i -4.2553e-01 + 0.0000e+00i 5.3755e-01 + 1.2279e-01i 2.5669e-01 + 0.0000e+00i -7.0577e-01 - 0.0000e+00i 6.9219e-01 + 0.0000e+00i D = Diagonal Matrix Columns 1 and 2: 1.0000e+00 + 0.0000e+00i 0 0 -2.3229e-01 + 9.3888e-02i 0 0 0 0 Columns 3 and 4: 0 0 0 0 -2.3229e-01 - 9.3888e-02i 0 0 2.8974e-01 + 0.0000e+00i octave:46> v=V(:,1) v = -3.9942e-01 -7.0088e-01 -4.9209e-01 -3.2722e-01 octave:47> invariantPotenze(P,1e-6,100) ans = 2.0807e-01 3.6512e-01 2.5635e-01 1.7046e-01 octave:48> v=v/sum(v) v = 2.0807e-01 3.6512e-01 2.5635e-01 1.7046e-01 octave:49> v*P error: operator *: nonconformant arguments (op1 is 4x1, op2 is 4x4) octave:49> P*v ans = 3.0227e-01 2.7200e-01 2.4569e-01 2.7096e-01 octave:50> v v = 2.0807e-01 3.6512e-01 2.5635e-01 1.7046e-01 octave:51> v=v' v = 2.0807e-01 3.6512e-01 2.5635e-01 1.7046e-01 octave:52> v*P ans = 2.0807e-01 3.6512e-01 2.5635e-01 1.7046e-01 octave:53> v v = 2.0807e-01 3.6512e-01 2.5635e-01 1.7046e-01 octave:54> P=randstoc(100); octave:55> [V,D]=eig(P'); octave:56> D(1,1) ans = 1.0000e+00 octave:57> V(:,1) ans = -1.0720e-01 -9.4778e-02 -9.5033e-02 -9.3382e-02 -1.0758e-01 -1.1550e-01 -1.0171e-01 -9.9766e-02 -9.9129e-02 -9.8895e-02 -8.8762e-02 -9.9619e-02 -1.0121e-01 -1.0310e-01 -1.0780e-01 -1.0238e-01 -9.5707e-02 -9.2890e-02 -9.4403e-02 -9.0603e-02 -1.0265e-01 -1.0549e-01 -8.9115e-02 -9.6275e-02 -9.8214e-02 -9.9391e-02 -8.9382e-02 -1.0044e-01 -1.0462e-01 -1.0143e-01 -9.9934e-02 -9.8931e-02 -1.1296e-01 -1.0239e-01 -1.0105e-01 -1.0794e-01 -1.1165e-01 -9.6035e-02 -1.0110e-01 -1.0887e-01 -9.8397e-02 -1.0343e-01 -9.2771e-02 -1.0213e-01 -9.3799e-02 -1.0674e-01 -1.0024e-01 -1.0044e-01 -8.8800e-02 -1.1156e-01 -9.9913e-02 -1.0037e-01 -1.1001e-01 -9.6005e-02 -9.4154e-02 -1.0290e-01 -8.8845e-02 -9.2794e-02 -1.0082e-01 -9.4018e-02 -9.8794e-02 -9.9326e-02 -1.0491e-01 -1.0556e-01 -9.4423e-02 -1.0320e-01 -9.9050e-02 -9.5905e-02 -9.7381e-02 -9.7683e-02 -1.1195e-01 -9.8713e-02 -9.5808e-02 -1.0395e-01 -1.0483e-01 -9.7344e-02 -8.8695e-02 -1.0970e-01 -9.6150e-02 -1.0186e-01 -9.2786e-02 -1.0232e-01 -9.6875e-02 -9.6972e-02 -9.7245e-02 -9.8717e-02 -1.0491e-01 -9.1860e-02 -9.4496e-02 -8.7636e-02 -9.4050e-02 -9.5406e-02 -1.1478e-01 -9.9076e-02 -1.0827e-01 -1.0029e-01 -1.0405e-01 -9.7339e-02 -1.0212e-01 -1.0153e-01 octave:58> octave:58> octave:58> octave:58> octave:58> octave:58> octave:58> octave:58> octave:58> octave:58> octave:58> octave:58> V(:,2) ans = -4.3830e-02 + 8.2620e-02i -2.1205e-02 + 4.3963e-02i -6.6694e-02 + 2.1953e-03i 4.0439e-02 - 8.6860e-02i 6.7734e-02 + 6.9579e-02i 6.7394e-03 - 4.9866e-03i 3.8171e-02 - 3.8659e-02i -1.0464e-01 + 6.6875e-03i -9.1259e-02 + 2.5250e-02i -1.2745e-01 + 8.4820e-02i 1.4393e-02 + 5.7770e-03i 1.0624e-01 + 9.6095e-02i -6.8784e-02 + 3.8116e-02i 1.0667e-01 - 1.0575e-02i 2.8006e-02 - 5.7317e-03i -1.3556e-02 - 2.0870e-01i -4.1374e-02 + 5.2677e-02i 6.4814e-02 - 1.9691e-02i -6.2465e-03 - 4.2118e-02i 1.1278e-01 - 1.5064e-01i 4.1122e-02 + 8.2995e-02i -8.0361e-02 - 9.5239e-03i -5.0354e-02 + 5.0892e-02i 1.0473e-01 + 8.1001e-02i -2.1823e-02 + 8.9211e-03i 1.7359e-03 - 1.1265e-01i 3.8454e-02 + 5.7582e-02i -2.5849e-02 + 1.6961e-01i -8.9854e-02 + 8.3696e-02i 5.5290e-02 - 2.5953e-02i -9.3177e-02 - 5.6037e-03i -7.9675e-02 - 2.4566e-03i 1.1843e-02 - 2.3681e-02i -4.0980e-02 - 1.1948e-02i -2.3857e-01 + 0.0000e+00i 8.5138e-02 - 3.7273e-02i -4.1504e-02 + 4.4052e-02i 8.7435e-02 - 1.2337e-02i -2.4005e-03 + 2.1581e-02i 8.9191e-02 - 2.2787e-02i 2.8785e-02 - 9.9421e-02i -1.3618e-01 + 3.6142e-02i 1.4986e-01 + 3.2201e-02i 9.6445e-02 + 3.8554e-02i -5.3149e-02 - 2.3982e-02i 2.6511e-02 - 9.6369e-02i -3.8375e-02 - 5.4963e-02i -4.1894e-02 - 1.0061e-01i 3.6498e-02 - 1.8260e-01i 1.0413e-01 - 1.2028e-01i 9.5977e-02 + 6.2791e-02i 5.1287e-02 + 1.1697e-01i -1.6378e-02 + 1.7857e-02i -2.4884e-02 + 6.7781e-02i -6.4181e-03 + 4.4843e-02i 6.0239e-02 + 5.7727e-02i -3.0770e-02 - 6.3748e-03i 2.0976e-02 + 7.8423e-02i -3.2094e-02 + 1.2547e-02i -7.1509e-03 - 3.4629e-02i 5.2955e-03 - 3.7051e-02i 2.8467e-02 - 2.9116e-02i 3.5213e-02 - 1.6699e-02i 3.7435e-02 - 1.0059e-01i -7.9855e-05 - 8.2111e-02i 5.8205e-02 + 3.9197e-02i -7.4033e-02 - 7.1698e-03i 4.2445e-02 - 4.4383e-02i 2.6193e-02 + 2.9341e-02i -5.5654e-04 + 8.0903e-02i -1.1026e-01 - 1.1764e-01i 8.4433e-02 - 1.3178e-02i 1.2305e-01 - 8.7410e-02i -9.9946e-02 - 5.9661e-02i 5.7543e-02 - 2.1300e-03i -2.8876e-02 - 2.2159e-04i 8.1647e-02 + 7.1407e-03i 4.0414e-02 + 5.3790e-03i -2.2432e-02 + 6.4478e-02i -1.9016e-01 + 2.6263e-03i 9.2093e-02 + 1.0807e-01i 1.0184e-03 + 1.2974e-01i -3.4095e-02 + 2.8206e-02i -1.3510e-01 + 6.2391e-02i 1.1302e-01 - 2.7784e-02i 3.9933e-02 + 3.8067e-03i -3.7632e-02 - 8.0685e-03i 4.5858e-02 + 5.0185e-02i -1.3466e-03 + 1.1837e-01i 2.1215e-02 + 5.1783e-02i -8.6021e-02 - 5.3897e-02i 4.5796e-02 - 2.7895e-02i -8.3803e-02 + 8.0810e-02i -7.6105e-02 - 8.0592e-02i 6.2569e-02 - 1.0841e-01i -3.5027e-02 + 3.2875e-02i -8.3636e-03 + 4.5456e-02i 1.3471e-02 - 1.9349e-01i -1.9821e-02 - 3.7884e-02i -4.6414e-02 + 7.2094e-02i octave:59> D(,1) parse error: syntax error >>> D(,1) ^ octave:59> D(1,1) ans = 1.0000e+00 octave:60> P=rand(4) P = 8.9796e-01 9.6313e-01 1.4734e-01 3.0007e-01 8.9709e-01 4.0521e-01 3.5136e-01 2.2649e-02 9.8826e-02 3.1774e-01 4.9868e-02 9.6832e-03 5.2033e-01 7.1185e-01 4.1019e-01 8.4274e-03 octave:61> (P-eye(4))'\zeros(4,1) ans = 0.0000e+00 0.0000e+00 -0.0000e+00 -0.0000e+00 octave:62> [(P-eye(4))';ones(1,4)]\[zeros(4,1);1] ans = 2.8956e-01 3.2845e-01 2.3687e-01 1.7908e-03 octave:63> v=[(P-eye(4))';ones(1,4)]\[zeros(4,1);1] v = 2.8956e-01 3.2845e-01 2.3687e-01 1.7908e-03 octave:64> v/sum(v) ans = 3.3801e-01 3.8341e-01 2.7649e-01 2.0904e-03 octave:65> ans' ans = 3.3801e-01 3.8341e-01 2.7649e-01 2.0904e-03 octave:66> ans*P ans = 6.7588e-01 5.7025e-01 1.9916e-01 1.1281e-01 octave:67> P=randstoc(4) P = 1.4385e-01 3.8064e-01 4.7262e-01 2.8922e-03 1.5684e-01 2.0430e-01 4.0266e-01 2.3619e-01 4.4140e-02 4.5886e-01 3.6042e-01 1.3659e-01 4.0082e-01 1.4282e-01 1.5439e-01 3.0197e-01 octave:68> v=[(P-eye(4))';ones(1,4)]\[zeros(4,1);1]; octave:69> v=v'/sum(v) v = 1.5762e-01 3.1169e-01 3.5509e-01 1.7560e-01 octave:70> v*P ans = 1.5762e-01 3.1169e-01 3.5509e-01 1.7560e-01 octave:71> S=randstoc(4) S = 1.4764e-01 2.6575e-01 1.2467e-01 4.6194e-01 5.3503e-02 2.0074e-01 3.9281e-01 3.5295e-01 1.4854e-01 1.3906e-01 5.6891e-01 1.4350e-01 2.3621e-02 3.1335e-01 5.1726e-01 1.4577e-01 octave:72> [Q,R]=qr(S'-eye(4)) Q = -8.4146e-01 1.5955e-02 2.0418e-01 5.0000e-01 2.6236e-01 -8.2154e-01 -7.9002e-02 5.0000e-01 1.2307e-01 4.1570e-01 -7.4970e-01 5.0000e-01 4.5603e-01 3.8988e-01 6.2452e-01 5.0000e-01 R = 1.0129e+00 -4.5412e-02 -7.6125e-02 -2.6356e-01 0.0000e+00 9.5838e-01 -2.3513e-01 -3.7508e-01 0.0000e+00 0.0000e+00 4.3215e-01 -9.4120e-01 0.0000e+00 0.0000e+00 0.0000e+00 4.4409e-16 octave:73> Q'*Q ans = 1.0000e+00 0.0000e+00 5.5511e-17 2.2204e-16 0.0000e+00 1.0000e+00 8.3267e-17 -5.5511e-17 5.5511e-17 8.3267e-17 1.0000e+00 5.5511e-17 2.2204e-16 -5.5511e-17 5.5511e-17 1.0000e+00 octave:74> Rtilde=R Rtilde = 1.0129e+00 -4.5412e-02 -7.6125e-02 -2.6356e-01 0.0000e+00 9.5838e-01 -2.3513e-01 -3.7508e-01 0.0000e+00 0.0000e+00 4.3215e-01 -9.4120e-01 0.0000e+00 0.0000e+00 0.0000e+00 4.4409e-16 octave:75> R(4,4)=1 R = 1.0129e+00 -4.5412e-02 -7.6125e-02 -2.6356e-01 0.0000e+00 9.5838e-01 -2.3513e-01 -3.7508e-01 0.0000e+00 0.0000e+00 4.3215e-01 -9.4120e-01 0.0000e+00 0.0000e+00 0.0000e+00 1.0000e+00 octave:76> Rtilde(4,4)=1 Rtilde = 1.0129e+00 -4.5412e-02 -7.6125e-02 -2.6356e-01 0.0000e+00 9.5838e-01 -2.3513e-01 -3.7508e-01 0.0000e+00 0.0000e+00 4.3215e-01 -9.4120e-01 0.0000e+00 0.0000e+00 0.0000e+00 1.0000e+00 octave:77> y=Rtilde\[0;0;0;1] y = 4.6537e-01 9.2570e-01 2.1779e+00 1.0000e+00 octave:78> y=y'/sum(y) y = 1.0185e-01 2.0260e-01 4.7668e-01 2.1887e-01 octave:79> y*S ans = 1.0185e-01 2.0260e-01 4.7668e-01 2.1887e-01 octave:80> diary off