Starting diary file "diary-2016-02-01.txt" on Mon Feb 01 2016 12:26 PM. A = randn(4) A = 0.5377 0.3188 3.5784 0.7254 1.8339 -1.3077 2.7694 -0.0631 -2.2588 -0.4336 -1.3499 0.7147 0.8622 0.3426 3.0349 -0.2050 help eig EIG Eigenvalues and eigenvectors. E = EIG(X) is a vector containing the eigenvalues of a square matrix X. [V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D. ... eig(A) ans = 0.0530 + 2.3375i 0.0530 - 2.3375i -1.2155 + 0.1047i -1.2155 - 0.1047i [V,D] = eig(A) V = Columns 1 through 2 0.6091 0.6091 0.4255 + 0.0454i 0.4255 - 0.0454i -0.2212 + 0.3827i -0.2212 - 0.3827i 0.4974 + 0.0551i 0.4974 - 0.0551i Columns 3 through 4 0.3241 + 0.0831i 0.3241 - 0.0831i -0.6582 -0.6582 -0.2230 - 0.0750i -0.2230 + 0.0750i 0.5940 + 0.2159i 0.5940 - 0.2159i D = Columns 1 through 2 0.0530 + 2.3375i 0 0 0.0530 - 2.3375i 0 0 0 0 Columns 3 through 4 0 0 0 0 -1.2155 + 0.1047i 0 0 -1.2155 - 0.1047i diag(D) ans = 0.0530 + 2.3375i 0.0530 - 2.3375i -1.2155 + 0.1047i -1.2155 - 0.1047i eig(A) ans = 0.0530 + 2.3375i 0.0530 - 2.3375i -1.2155 + 0.1047i -1.2155 - 0.1047i A*V ans = Columns 1 through 2 0.0323 + 1.4238i 0.0323 - 1.4238i -0.0834 + 0.9970i -0.0834 - 0.9970i -0.9062 - 0.4968i -0.9062 + 0.4968i -0.1024 + 1.1656i -0.1024 - 1.1656i Columns 3 through 4 -0.4026 - 0.0671i -0.4026 + 0.0671i 0.8000 - 0.0689i 0.8000 + 0.0689i 0.2789 + 0.0678i 0.2789 - 0.0678i -0.7446 - 0.2003i -0.7446 + 0.2003i V*D ans = Columns 1 through 2 0.0323 + 1.4238i 0.0323 - 1.4238i -0.0834 + 0.9970i -0.0834 - 0.9970i -0.9062 - 0.4968i -0.9062 + 0.4968i -0.1024 + 1.1656i -0.1024 - 1.1656i Columns 3 through 4 -0.4026 - 0.0671i -0.4026 + 0.0671i 0.8000 - 0.0689i 0.8000 + 0.0689i 0.2789 + 0.0678i 0.2789 - 0.0678i -0.7446 - 0.2003i -0.7446 + 0.2003i A*V - V*D ans = 1.0e-14 * Columns 1 through 2 -0.1395 + 0.0222i -0.1395 - 0.0222i -0.2345 + 0.1332i -0.2345 - 0.1332i 0.1332 + 0.0500i 0.1332 - 0.0500i -0.1388 + 0.0222i -0.1388 - 0.0222i Columns 3 through 4 -0.0222 - 0.0194i -0.0222 + 0.0194i -0.1221 - 0.0389i -0.1221 + 0.0389i 0.0555 + 0.0264i 0.0555 - 0.0264i -0.0333 - 0.0194i -0.0333 + 0.0194i A A = 0.5377 0.3188 3.5784 0.7254 1.8339 -1.3077 2.7694 -0.0631 -2.2588 -0.4336 -1.3499 0.7147 0.8622 0.3426 3.0349 -0.2050 At = A' At = 0.5377 1.8339 -2.2588 0.8622 0.3188 -1.3077 -0.4336 0.3426 3.5784 2.7694 -1.3499 3.0349 0.7254 -0.0631 0.7147 -0.2050 eig(At) ans = 0.0530 + 2.3375i 0.0530 - 2.3375i -1.2155 + 0.1047i -1.2155 - 0.1047i eig(A) ans = 0.0530 + 2.3375i 0.0530 - 2.3375i -1.2155 + 0.1047i -1.2155 - 0.1047i [VA,DA] = eig(A) ... [VAt,DAt] = eig(At) ... A = A + A' A = 1.0753 2.1527 1.3196 1.5876 2.1527 -2.6154 2.3358 0.2796 1.3196 2.3358 -2.6998 3.7497 1.5876 0.2796 3.7497 -0.4099 [V,D] = eig(A) V = -0.1072 0.3982 -0.6892 0.5958 0.4798 -0.7752 -0.2254 0.3437 -0.7356 -0.3392 0.3347 0.4815 0.4660 0.3543 0.6018 0.5433 D = -6.4063 0 0 0 0 -2.8269 0 0 0 0 -0.2476 0 0 0 0 4.8311 clear clc load pagerankmats whos Name Size Bytes Class Attributes EG1 4x4 128 double EG2 5x5 200 double EG3 500x500 2000000 double HarvardCrawl 500x1 96816 cell E = EG1 E = 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 [n,n] = size(E) n = 4 n = 4 nnz(E) ans = 8 outdegree = sum(E) outdegree = 3 2 1 2 indegree = sum(E') indegree = 2 1 3 2 E * ones(n,1) ans = 2 1 3 2 x = indegree x = 2 1 3 2 x = x/norm(x) x = 0.4714 0.2357 0.7071 0.4714 norm(x) ans = 1.0000 [xx, perm] = sort(x) xx = 0.2357 0.4714 0.4714 0.7071 perm = 2 1 4 3 [xx, perm] = sort(x,'descend') xx = 0.7071 0.4714 0.4714 0.2357 perm = 3 1 4 2 E E = 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 eig(E) ans = 1.9498 -0.7454 + 0.7495i -0.7454 - 0.7495i -0.4590 sum(E) ans = 3 2 1 2 A = E * diag( 1 ./ sum(E)) A = 0 0 1.0000 0.5000 0.3333 0 0 0 0.3333 0.5000 0 0.5000 0.3333 0.5000 0 0 eig(A) ans = 1.0000 -0.3606 + 0.4110i -0.3606 - 0.4110i -0.2788 A' ans = 0 0.3333 0.3333 0.3333 0 0 0.5000 0.5000 1.0000 0 0 0 0.5000 0 0.5000 0 sum(A) ans = 1 1 1 1 A' * ones(4,1) ans = 1 1 1 1 A' * ones(4,1) ans = 1 1 1 1 A * ones(4,1) ans = 1.5000 0.3333 1.3333 0.8333 x x = 0.4714 0.2357 0.7071 0.4714 [xx, perm] = sort(x,'descend') xx = 0.7071 0.4714 0.4714 0.2357 perm = 3 1 4 2 [V,D] = eig(A) V = Columns 1 through 2 0.7210 0.7552 0.2403 -0.3037 - 0.3461i 0.5408 -0.0932 + 0.2747i 0.3605 -0.3584 + 0.0714i Columns 3 through 4 0.7552 0.5065 -0.3037 + 0.3461i -0.6057 -0.0932 - 0.2747i -0.3815 -0.3584 - 0.0714i 0.4807 D = Columns 1 through 2 1.0000 0 0 -0.3606 + 0.4110i 0 0 0 0 Columns 3 through 4 0 0 0 0 -0.3606 - 0.4110i 0 0 -0.2788 D(1,1) ans = 1.0000 ans - 1 ans = 1.1102e-15 x = V(:,1) x = 0.7210 0.2403 0.5408 0.3605 norm(x) ans = 1 perm perm = 3 1 4 2 [xx, perm] = sort(x,'descend') xx = 0.7210 0.5408 0.3605 0.2403 perm = 1 3 4 2 E E = 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 E(1,3) = 0 E = 0 0 0 1 1 0 0 0 1 1 0 1 1 1 0 0 A = E * diag( 1 ./ sum(E)) A = 0 0 NaN 0.5000 0.3333 0 NaN 0 0.3333 0.5000 NaN 0.5000 0.3333 0.5000 NaN 0 eig(A) {??? Error using ==> eig Input to EIG must not contain NaN or Inf. } E E = 0 0 0 1 1 0 0 0 1 1 0 1 1 1 0 0 outdegree = sum(E) outdegree = 3 2 0 2 for j=1:n, if outdegree(j)==0, E(:,j) = ones(n,1); end;end E E = 0 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 A = E * diag( 1 ./ sum(E)) A = 0 0 0.2500 0.5000 0.3333 0 0.2500 0 0.3333 0.5000 0.2500 0.5000 0.3333 0.5000 0.2500 0 [V,D] = eig(A) V = Columns 1 through 2 0.4143 0.5979 0.3157 -0.2617 - 0.5000i 0.7103 0.0819 + 0.3435i 0.4735 -0.4181 + 0.1565i Columns 3 through 4 0.5979 0.5979 -0.2617 + 0.5000i -0.0746 0.0819 - 0.3435i -0.7617 -0.4181 - 0.1565i 0.2383 D = Columns 1 through 2 1.0000 0 0 -0.3154 + 0.2745i 0 0 0 0 Columns 3 through 4 0 0 0 0 -0.3154 - 0.2745i 0 0 -0.1192 x = V(:,1) x = 0.4143 0.3157 0.7103 0.4735 [xx, perm] = sort(x,'descend') xx = 0.7103 0.4735 0.4143 0.3157 perm = 3 4 1 2 diary off