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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J. M.'s missing motivation
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Mencia
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You are talking about matrices with numbers as elements, right? – halirutan Jan 21 '14 at 14:34
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@halirutan , yes indeed, complex numbers in general. – Mencia Jan 21 '14 at 14:35
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In the case where it is positive definite, possibly diagonalizing, via eigensystem, a Cholesky factor might give a speed boost. – Daniel Lichtblau Jan 21 '14 at 15:33
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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} *)
J. M.'s missing motivation
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ybeltukov
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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) – L.K. Jan 31 '17 at 15:58
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1@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. – ybeltukov Jan 31 '17 at 16:27