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Eigenvectors
12-27-2018, 12:58 PM (This post was last modified: 12-27-2018 01:08 PM by parisse.)
Post: #2
RE: Eigenvectors
That's because the algorithms are not the same. If you compute the eigenvectors of an approx matrix, the eigenvectors are deduced from the Schur factorization SCHUR(m), not by factorizing the characteristic polynomial. The first matrix P returned by SCHUR is unitary, it's first column is an eigenvector with euclidean norm 1, the second eigenvector is deduced by a linear combination of both column vectors of P to cancel the non diagonal coefficient of the second matrix returned by SCHUR (and it is not normalized).
If there is something to improve here mathematically speaking, it's certainly not to convert to exact like Wolfram Alpha does. It would be to normalize the first eigenvector (the eigenvector corresponding to eigenvalue 1) with respect to the L1 norm (sum of absolute values), to get the invariant probability vector (2/3.,1/3.) (you can check it by powering your matrix to a high power, like m^100), that you can also get by solving m*v=v if you know that m is a stochastic matrix (the best way to do that is to iterate if m is approx).
This also demonstrates that having too much confidence in Wolfram Alpha is not a good idea, it may look simpler, but it's not always appropriate (this is of course true for any CAS, but more true for those with an interface where everything is supposed to be done for you)
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Messages In This Thread
Eigenvectors - DrD - 12-27-2018, 11:40 AM
RE: Eigenvectors - parisse - 12-27-2018 12:58 PM
RE: Eigenvectors - DrD - 12-27-2018, 03:58 PM
RE: Eigenvectors - parisse - 12-27-2018, 04:32 PM
RE: Eigenvectors - jte - 08-20-2021, 08:21 PM
RE: Eigenvectors - John Keith - 12-27-2018, 04:42 PM
RE: Eigenvectors - compsystems - 12-27-2018, 04:50 PM
RE: Eigenvectors - parisse - 12-27-2018, 04:51 PM
RE: Eigenvectors - compsystems - 12-27-2018, 05:30 PM
RE: Eigenvectors - rawi - 08-21-2021, 07:48 AM



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