I Lecture notes on SUSY using finite matrix as example?

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arivero
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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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Maybe try SUSY transformation and Cholesky factorization.
 
https://arxiv.org/pdf/2503.09804 From the abstract: ... Our derivation uses both EE and the Newtonian approximation of EE in Part I, to describe semi-classically in Part II the advection of DM, created at the level of the universe, into galaxies and clusters thereof. This advection happens proportional with their own classically generated gravitational field g, due to self-interaction of the gravitational field. It is based on the universal formula ρD =λgg′2 for the densityρ D of DM...
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