Obtain the largest positive eigenvalue (with its eigenvector)?

Question

Eigenvalues[M,1] can be used to return the largest eigenvalue in absolute value. Is there a simple way to obtain the largest positive eigenvalue instead, as well as the corresponding eigenvector/s?

@kglr That gives the largest eigenvalue according to absolute value. I want the largest positive eigenvalue (there might be another negative eigenvalue with larger absolute value that I do not want). — a06e, Jul 07 '17 at 13:33

kglr · Answer 1 · 2017-07-07T14:27:30.037

2

evv = Module[{es = Eigensystem[#], ord}, ord = Ordering[-es[[1]]]; es[[All, ord[[1]]]]] &;


m = N[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}];    
evv @ m

{16.1168, {-0.231971, -0.525322, -0.818673}}

evv[-m]

{1.11684, {0.78583, 0.0867513, -0.612328}}

Update: Using the options in Jens's answer in the q/a linked by @Carl

evv2 = Eigensystem[#, 1, Method -> {"Arnoldi", "Criteria" -> "RealPart"}][[All,1]]&;
evv2@ m

{16.1168, {-0.231971, -0.525322, -0.818673}}

edited Jul 07 '17 at 14:27

answered Jul 07 '17 at 13:39

kglr

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There is no way to bypass the computation of all the eigenvalues when you are only interested in the largest positive one? – a06e Jul 07 '17 at 14:04
@becko, good point. i can't think of any way. – kglr Jul 07 '17 at 14:11
I figured out something that could work in some cases, see my answer. – a06e Jul 07 '17 at 14:15

TimRias · Answer 2 · 2017-07-07T14:06:42.707

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This should do it:

Last@Sort@Cases[Transpose@Eigensystem@M,{_Real,__}]

edited Jul 07 '17 at 14:06

answered Jul 07 '17 at 13:57

TimRias

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It would be helpful to demonstrate the code by running in on m given in other answers. – bbgodfrey Jul 07 '17 at 14:41

Obtain the largest positive eigenvalue (with its eigenvector)?

2 Answers2