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I'm working on helping my friend create code to perform the Overlap Dirac Operator and have come across one part that I'm not sure what to do.

I need to compute the Eigenvalues and corresponding Eigenvectors for a hermitian matrix (my test is for n=4) so that I can implement a matrix sign function $epsilon(M) = U * epsilon(A) * U^*$ where $U$ is an eigenvector matrix, $U^*$ is the complex transpose of $U$, and $epsilon(A)$ is a matrix with the signs of eigenvalues of $M$ on it's diagonal.

Any resources or help on existing algorithms/libraries to solve this (or at least make close approximations) would be appreciated. My target programming language is Scala.

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Welcome to Computational Science Phillip. I've updated your question with the MathJax mark-up I think you intended, but could you check to make sure I haven't missinterpreted what you meant. –  Mark Booth Jan 25 '13 at 0:52
I think the description of $epsilon(A)$ as vector $lambda * I$ is not correct, since this would inevitably be vector $lambda$. Instead something like a diagonal matrix with those signed entries is probably meant. –  hardmath Jan 25 '13 at 3:14
I think that my description might've just been poor to begin with. I'm new to this material. What I mean is that $epsilon(M) = U*epsilon(A)*U^*$ where $epsilon(A)$ is a matrix with the signs of eigenvalues of $M$ on the diagonal. –  Phillip Huff Jan 25 '13 at 3:51
The subroutine you are looking for should work only for $n=4$ or also for $n>4$? –  Stefano M Jan 25 '13 at 9:13
Why aren't you just computing the eigenvalue decomposition and then set $\epsilon(A)=diag(sign\lambda_i)$ where $\lambda_i$ are the eigenvalues? –  Deathbreath Jan 25 '13 at 14:15

1 Answer 1

Sorry I delayed so long to answer this question, but I wanted to answer this question in case someone else has a problem

I found a solution to this problem by using the jBLAS library (http://mikiobraun.github.io/jblas/)

To calculate the sign function $epsilon(M) = U * epsilon(A) * U^\dagger$, if you have an existing matrix matrixR in jBlas:

val eigenvecs = Eigen.eigenvectors(matrixR)
val lambda = new DoubleMatrix(4,4, math.signum(eigenvecs(1).get(0,0).real),0,0,0,
  0,0,0,math.signum(eigenvecs(1).get(3,3).real)) //can be extended to larger matrices
val epsilon = eigenvecs(0).real.mmul(lambda).mmul(eigenvecs(0).real.transpose) //where epsilon = epsilon(M)
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