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An exercise on the manifold of derogatory matrices

17 Novembre 2013, 18:21pm

Publié par Fabien Besnard

Here is a nice little exercise. Let us consider the set H_N of all hermitian NxN matrices. In this circumstance, a matrix M is called derogatory if it has at least one multiple eigenvalue (in more general circumstances the definition is more complex, but never mind). In fact we are interested in the set A of hermitian matrices which are block diagonal, with the size (and place) of the blocks fixed. In other words we are looking at the direct sum A of H_{n_1},...,H_{n_k} with n_1+...+n_k=N.


We suppose that k>1.


Let us call V the manifold of derogatory matrices inside A. It decomposes into different components, according to whether the multiplicities fall inside the same block or not. We are interested in the highest dimensional components, which have dimension N-1 (why ?) and correspond to the case where the multiplicities are across different blocks (say the first two for concreteness).


The question is : what is the equation of the tangent hyperplane to V at some matrix a the two first blocks of which share an eigenvalue ?


I solved this in a rather heuristic way, guessing the answer from a simple example, and then checking that it holds generally. I wonder if there is a more systematic ( but still simple) way to derive the result.


I'll give the answer at the end of the week.



Update 20/11/2013 : Since nobody gave the answer, here is an indication. You might like to first work out the case where the blocks have all size 1.