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Behavior of Eigenvalues and Eigensystem

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  • Subject: [mg115072] Behavior of Eigenvalues and Eigensystem
  • From: "Eric Michielssen" <emichiel at>
  • Date: Thu, 30 Dec 2010 04:11:10 -0500 (EST)


I am running into some trouble using Eigenvalues and Eigensystem.

I run 

Eigensystem[mat1,mat2,3, Method -> Arnoldi] 

where mat1 and mat2 are both sparse, Hermitian, positive definite matrices.
Mathematica confirms this: when I run HermitianMatrixQ[mat1] or
PositiveDefiniteMatrixQ[mat1] both return True, same for mat2. I also
explicitly compute these matrices' eigenvalues, and they all are positive.

Yet, when I run the Eigensystem command then one of two things happens:

1. Mathematica complains that the matrices are not Hermitian. OK, they
sometimes are ever so slightly non Hermitian, say by having an imaginary
component 10^(-21) on the main diagonal. I can eliminate that by
Chopping/symmetrizing, but that comes at the cost of additional memory, and
mat1 and mat2 can be huge.

2. Even when I do so, Mathematica complains that the second matrix is not
positive definite:

Eigensystem::chnpdef: The second matrix
33\[VeryThinSpace]+0. I,-0.666667+0. I,-2.81637+0.719138 I,-0.666667+0.
I,4.33333\[VeryThinSpace]+0. I,-2.81637+0.719138 I,-2.81637-0.719138
I,4.33333\[VeryThinSpace]+0. I,-0.666667+0. I,-2.81637-0.719138
I,-0.666667+0. I,4.33333\[VeryThinSpace]+0. I}}]
 in the first argument seems not to be positive definite, which is required
for the Arnoldi method. >>


Note: I use Method -> Arnoldi as Mathematica suggests I do so as my matrices
are highly sparse, and huge. I cannot find any documentation on Method ->
Arnoldi, though. Are there other dials that can be set when using this
option? Are there other options available for Eigenvalues/system?

Eric Michielssen

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