Square roots of nilpotent matrices

Oddly enough, it seems that after all the discussion at this post, no-one actually wrote down a criterion for a nilpotent matrix to have a square root.

The answer is very nice, in fact:

Proposition. A nilpotent matrix has a square root if and only if the (2n-1)-th Jordan block (from largest to smallest) is the same size, or one larger than 2n-th.

Who feels like giving a proof in comments?

EDIT: I originally bollixed the condition, by transposing the partition given by the Jordan blocks (the above condition is the same as the partition given by the Jordan blocks having transpose where each odd row length appears at most once).

11 thoughts on “Square roots of nilpotent matrices

  1. If I’m not mistaken, the square of a nilpotent 2n-block is two n-blocks, while the square of a (2n+1)-block is an n-block and an (n+1)-block. You can then order the blocks by decreasing size, separate into pairs, and note that admissible pairs have size off by at most one. We can eliminate the boundary case of an odd number of blocks by allowing blocks of size zero.

  2. I did give the condition in comment #17: “The nilpotent term has a square root iff Jordan blocks can be paired so that the two blocks in each pair are either the same size or adjacent sizes.”

    The proof follows directly from the Jordan canonical form of the square root.

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