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Aug 20, 2019 at 19:34 comment added Gerhard Paseman @Durzot, indeed, but I wanted to paint the idea with broad strokes and almost no notation. Once you have the idea of the structure, you should have little problem writing it. You can go for a non inductive presentation as well, but this I find easier to communicate. Gerhard "It's All About The Understanding" Paseman, 2019.08.20.
Aug 20, 2019 at 19:07 comment added LSpice The induction is even easier to start at $d = 0$.
Aug 20, 2019 at 19:07 comment added Durzot Thank your for your answer. If I understand correctly you rewrite the matrix in a permutation of the canonical basis for $\mathbb{R}^{rd+r}$ so as to make $B$ appear in the upper left corner of size $r\times r$. If that is the case your idea is very similar to the idea in the post indicated by Robert Israel where no recurrence is needed.
Aug 20, 2019 at 17:07 history answered Gerhard Paseman CC BY-SA 4.0