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Perturbation of multiple eigenvalues

For the past several months my Saturdays consist of two meetings, each of which about 4 hours long, with two of my colleagues working on various problems. Clearly we’ve hit a lot of dead ends, and we’ve got several interesting results too. Today’s first meeting specially was interesting because it boiled down to an understanding that one of our conjectures is not correct. The fact that the conjecture is not correct was clear from a counterexample, but the understanding was established from a point in our method to prove the conjecture.

The conjecture deals with the multiplicity of eigenvalues of a matrix with a fixed graph, and the idea is to use the Jacobian method with a nice starting graph. Everything works perfectly, and at first it seemed too good to be true (which actually was too good to be true) but then we modified it a lot to get to a point that it was good enough to be believable. But then we had this counterexample. Nonetheless, we carried our method and proved the (wrong) conjecture, then went back and analysed it until we precisely found the gaffe.

That happens quite often, but the interesting part is that the gaffe seems to be fixable in certain cases, and that will probably lead us to an interesting result, and better understanding of basic matrix analysis about matrix perturbation, multiplicity of eigenvalues, and the corresponding eigenspaces

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