Is there any faster way than using Eigensystem to diagonalize (get all the eigenvectors and eigenvalues) of a Hermitian (self-adjoint) matrix?
That would be amazing :).
Thanks.
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$\begingroup$ You are talking about matrices with numbers as elements, right? $\endgroup$– halirutanCommented Jan 21, 2014 at 14:34
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$\begingroup$ @halirutan , yes indeed, complex numbers in general. $\endgroup$– MenciaCommented Jan 21, 2014 at 14:35
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$\begingroup$ In the case where it is positive definite, possibly diagonalizing, via eigensystem, a Cholesky factor might give a speed boost. $\endgroup$– Daniel LichtblauCommented Jan 21, 2014 at 15:33
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1 Answer
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I think the answer is no. Eigensystem already uses faster algorithms for Hermitian matrices. See what happens when I add a small non-Hermitian matrix:
n = 1000;
m = RandomComplex[1 + I, {n, n}];
h = m + ConjugateTranspose[m];
d = 10^-10 RandomComplex[1 + I, {n, n}];
Eigensystem[h]; // AbsoluteTiming
(* {2.971269, Null} *)
Eigensystem[h + d]; // AbsoluteTiming
(* {14.567275, Null} *)
answered Jan 21, 2014 at 21:04
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$\begingroup$ A query after your helpful answer. Is there a chance where Mathematica assumes Matrix not to be Hermitian, even though it is Hermitian?(This is what happened with me) $\endgroup$– L.K.Commented Jan 31, 2017 at 15:58
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1$\begingroup$ @L.K. Your matrix can be Hermitian up to some numerical precision depending on your previous computations. You can make it Hermitian in a strict sense with
h = (h + ConjugateTranspose[h])/2. $\endgroup$ Commented Jan 31, 2017 at 16:27