Normalization of complex eigenvector
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AVM
el 1 de Feb. de 2020
What is the best way of normalisation of a complex eigenvector of a complex hermitian matrix. Here I am doing this in the following way, but norm remaim as it is.
syms theta phi
a=[cos(theta) sin(theta)*exp(1i*phi);sin(theta)*exp(-1i*phi) cos(theta)];
[V,~]=eig(a);
V(:,1)/norm(V(:,1))
This produces the vector as
exp(phi*1i)/(exp(-2*imag(phi)) + 1)^(1/2)
1/(exp(-2*imag(phi)) + 1)^(1/2)
But Normalization factor remain in symbolic form, but it should by sqrt(2). Pl somebody help me to understand.
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Vladimir Sovkov
el 1 de Feb. de 2020
You probably want
V(:,1) = V(:,1)/norm(V(:,1));
Besides, if your theta and phi are supposed to be real, the overall computation would be simpler with the assumptions
syms theta phi
assume(theta,'real');
assume(phi,'real');
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Walter Roberson
el 3 de Feb. de 2020
Well, given sufficient time you can get some result, but if you want values that mean anything, you need to evaluate symbolically to 80 or so digits and set a relatively large numeric integration target.
If you are willing to give up that approximately 2% contribution term then the process would be to extract the 3rd term of r, evaluate it at a matrix of theta and k1, evalf(), and integrate the matrix over phi = 0..2*pi and the integration will not take a grindingly long time. If you keep the approximately 2% contribution term you would probably have to set a larger permitted relative error term in order to prevent the numeric integration from being very very slow.
Integrating r as a whole and then substituting k1 and theta into it is probably going to take much too long.
If you substitute in a particular theta value then the third term of r expands to a surprisingly long expression involving k1 and phi that is not profitable to do an exact integration on with respect to phi.
I don't know if an exact integral of r even exists. I had to kill the exact integration when it got up to 75 gigabytes of memory on my system.
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