Weyr characteristic

For a fixed eigenvalue lambda there are two equivalent ways to record its Jordan data. The block-size partition lists how big the blocks are. The Weyr characteristic lists, instead, how many blocks have size at least k, for k = 1, 2, 3, .... These two lists are conjugate (transpose) partitions — the same information read along rows versus columns.

Concretely, set w_k = dim ker(N^k) - dim ker(N^{k-1}) where N = A - lambda*I (so w_k = the first differences of the nullities). The sequence w_1 >= w_2 >= w_3 >= ... is the Weyr characteristic of lambda. Here w_k counts the Jordan blocks of size at least k, and the largest index with w_k > 0 is the size of the biggest block.

The Weyr and Segre (block-size) partitions are conjugate: if you draw the block sizes as rows of a Young diagram, the Weyr numbers are its column heights, and vice versa. So either one determines the other; the Weyr form is just the more rank-friendly bookkeeping, since each w_k is a clean dimension difference you compute directly.

Why prefer Weyr: its entries come straight from kernel dimensions of matrix powers, so it is what the rank-jump formula actually outputs first. It also generalizes to the Weyr canonical form, an alternative to Jordan that is nicer for problems about commuting matrices.