complex eigenvalues

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zayed 2011-12-5

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Hi, I have a square symmetric matrix (5,5) with complex entries,the output eigenvalues when I use eig(T) are all complex .I want to determine the smallest negative eigenvalue.I don't know how ,any one can help.

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Sean de Wolski 2011-12-5

smallest as in the closest to infinity or closest to zero?

zayed 2011-12-5

smallest is the minimum one,so if it's negative it will be closest to minus infinity,if it's positive it will be closest to zero.

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Walter Roberson 2011-12-5

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"smallest" is not defined for complex numbers. "negative" is not defined for complex numbers either.

You can compare real parts, or you can compare imaginary parts, or you can compare magnitudes.

[vals, idx] = min(real(E));
E(idx)

[vals, idx] = min(imag(E));
E(idx)

[vals, idx] = min(abs(E));
E(idx)

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Walter Roberson 2011-12-6

hypot has been in versions since sometime in 2008 or before; I have not traced it further.

When I read Loren's blog about hypot, I see in the comments that abs() is also implemented robustly, so there would be no advantage to using hypot() over using abs(), so you might as well not bother.

http://blogs.mathworks.com/loren/2008/02/07/why-hypot/

If you are looking for the eigenvalue with the smallest magnitude (such as min(abs(E)) would find), then you could instead use

gamma = eigs(T,1,'sm');

which will find just the one eigenvalue. Smallest magnitude could be positive or negative for the real or imaginary components, though -- the eigenvalue closest to 0. There is unfortunately no way with eigs to pick out just the complex eigenvalue with the real component or imaginary component closest to negative infinity: you will have to use one of the above min() forms for that.

zayed 2011-12-6