Page 63 - Textos de Matemática Vol. 44

A PASCAL TYPE MATRIX AND THE TENSOR PRODUCT OF JORDAN BLOCKS
ROD GOW AND THOMAS J. LAFFEY
Dedicated to Nat´alia Bebiano on her birthday
Abstract. The factorization of a Pascal-type matrix P of binomial coeffi- cients modulo a prime p in the form LQU, with L lower triangular, U upper triangular and Q a permutation matrix is studied. A surprising connection is found between the form of Q and the Jordan form of tensor products of Jordan blocks in characteristic p.
1. Result
Let F be a field. For 0 ̸= a ∈ F, the n×n full Jordan block Jn(a) with eigenvalue a is similar to aJn(1), so, in describing the Jordan canonical form of Jm(a) ⊗ Jn(b), for a, b ∈ F with ab ̸= 0, it suffices to determine the Jordan form of Jm(1) ⊗ Jn(1).
Given a positive integer r, we write Ir for the r × r identity matrix and Jr for Jr(1).
An elementary but key observation is that the matrix
is equivalent to and also to
xImn −Jm ⊗Jn Im(n−1) ⊕ (xIm − Jm)n
I(m−1)n ⊕ (xIn − Jn)m
via elements of GL(mn, Z[x]), where Z denotes the ring of integers. To see this,
notethatinthecasen=2,theKroneckerformofxImn−Jm⊗Jn is xIm − Jm −Jm
2010 Mathematics Subject Classification. 15A21, 15A23, 20C20
Key words and phrases. Jordan canonical form, tensor product decomposition, finite charac- teristic, matrix factorization
53
R := 0 xIm − Jm