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Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?

I guess it has to be its Jordan normal form but I am not sure.

Remarks:

  • A matrix is sparser than other if it has less nonzero entries.

  • Two square $n \times n$ matrices $A,C$ are similar if there exists and invertible matrix $P$ such that $A = P^{-1}CP$

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    $\begingroup$ Simulposted to (and answered at) m.se, math.stackexchange.com/questions/2961899/… $\endgroup$ – Gerry Myerson Oct 19 '18 at 12:00
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    $\begingroup$ Setting aside the cross-posting, the MSE answer shows that the guess is wrong but doesn't answer the question (which I assume is hard). $\endgroup$ – Ben Barber Oct 19 '18 at 13:58

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