Perron eigenvector
in [Matlab]
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From: d on 23 May 2010 13:55 hi, Is there's a matlab function that calculates the Perron vector of a matrix ? (the eigenvector that corresponds to the "Perron root" or the "Perron–Frobenius eigenvalue") ? thank you, D
From: d on 23 May 2010 14:21 just to make it clear - I wish to use it on a non-negative matrix. (of course, for a strictly positive matrix A, [e,v] = eigs(A,1) would do) thanks again, D
From: Matt J on 23 May 2010 15:07 "d " <dagkatan(a)yahoo.com> wrote in message <htbq5q$3ka$1(a)fred.mathworks.com>... > hi, > Is there's a matlab function that calculates the Perron vector of a matrix ? > (the eigenvector that corresponds to the "Perron root" or the "Perron–Frobenius eigenvalue") ? ============= Something like this perhaps: [V,D]=eig(myMatrix); [trash,idx]=max(abs(diag(D))); PerronVector=V(:,idx);
From: Bruno Luong on 23 May 2010 15:56 "Matt J " <mattjacREMOVE(a)THISieee.spam> wrote in message <htbuco$59a$1(a)fred.mathworks.com>... > "d " <dagkatan(a)yahoo.com> wrote in message <htbq5q$3ka$1(a)fred.mathworks.com>... > > hi, > > Is there's a matlab function that calculates the Perron vector of a matrix ? > > (the eigenvector that corresponds to the "Perron root" or the "Perron–Frobenius eigenvalue") ? > ============= > > Something like this perhaps: > > [V,D]=eig(myMatrix); > [trash,idx]=max(abs(diag(D))); > PerronVector=V(:,idx); Not quite, first we have to note that using max(diag(D)) is correct. But even with that change, I don't believe there is any warranty the above would return Perron vector. For example let Q=[0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0]; We have V1 = [1; 1; 0; 0] V2 = [0; 0; 1; 1] are eigen vectors corresponding to rho(Q)=1, and both are Perron vectors. However there is no thing to prevent Matlab to return V1 = [1; 1; -1; -1] V2 = [2; 2; -1; -1] for example. Both of course are not Perron's vector. That's where the difficulty arises. Bruno
From: Bruno Luong on 23 May 2010 16:39 I suggest to use LINPROG (opt toolbox) to eventually "fix" the eigen vector. [V D]=eig(M); lambda = diag(D); rho = max(lambda); tol = 1e-10; % use some small tolerance ind = find(lambda>=rho*(1-tol)); V = V(:,ind); % Use linear programming to find a combination of V that % has non-negative elements % Vperron := V*x % Vperron >= 0 <=> -V*x w= 0 % sum(V*x) = 1 <=> sum(V,1)*x = 1 A = -V; b = zeros(size(V,1),1); Aeq = sum(V,1); beq = 1; dummy = ones(size(V,2),1); x = linprog(dummy, A, b, Aeq, beq); Vperron = V*x % Bruno
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