Lecture notes on SUSY using finite matrix as example?

In summary, the factorisation trick is a key concept in many SUSY examples, particularly in Witten's "SUSY QM". This involves producing two Hamiltonians, H0=AA+ and H1=A+A, and showing that they have the same eigenvalues except for zeros. The typical proof involves considering an eigenvector of H0, |Ψ>, and using the equation H1|A+Ψ>=λ|A+Ψ>, where λ is an eigenvalue of H0. This idea can be illustrated with a 2x3 matrix, but it may also be found in other sources such as lecture notes or blog entries. However, combining the search terms "susy lectures" and "
  • #1
arivero
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TL;DR Summary
Eigen(AA^+)=Eigen(A^+A)
As you know, a lot of SUSY examples, particularly from Witten's "SUSY QM", pivot on the factorisation trick: produce two hamiltonians $$H_0=AA^+, H_1=A^+A$$ and see they have the same eigenvalues except for ceros.

The proof usually goes by: let ##\Psi## be an eigenvector of ##H_0##, consider ##|A^+\Psi \rangle ##, then $$H_1 |A^+\Psi \rangle =A^+AA^+\Psi=A^+H_0\Psi=A^+\lambda\Psi=\lambda|A^+\Psi \rangle$$
Now this can be already seen if A a 2x3 matrix, and I think that I have sometimes this example as an starting point but just now I can not locate it, do any of you remember perhaps a blog entry or, better, any set of lecture notes doing this? With finite matrices, I mean.

I got the idea of searching simultaneously for "susy lectures" and "Cholesky factorisation" but no results.
 
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  • #2
Maybe try SUSY transformation and Cholesky factorization.
 

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