Page 65 - Textos de Matemática Vol. 44

Suppose that
A PASCAL-TYPE MATRIX 55
Pn,m = LQU,
where Q = (qij) is a permutation matrix and L and U are lower and upper triangular matrices, repectively, over F.
Then the tensor product Jm ⊗ Jn over F has similarity invariants (x − 1)w(i,j)
where the positions (i, j) are those for which qij = 1. Proof. Let Pn,m(x) be the m × m matrix
 n xn−m+1 n xn−m+2 m−1 m−2
 n xn−m+2 n xn−m+3 m−2 m−3
· · ·
· · ·
n xn−2 n xn−1 n xn 2 1 0
n xn−1 n xn 0  1 0 
...... . .......
 n xn−1 n xn ··· 0 0 0  10
 n xn 0 ··· 0 0 0 0
Let V = (vij) be the m×m reverse identity matrix; that is vm1 = ··· = v1m = 1, and vij = 0, for all other (i, j ). Note that the matrices (xIm − Jm )n and Pn,m(x − 1)V are equivalent since
(xIm − Jm)n = (−1)iδij Pn,m(x − 1)V (−1)iδij . m×m m×m
Hence the Smith invariants of (xIm − Jm)n are the same as those of Pm,n(x) with x replaced by x − 1.
Next,letY =(yij)beanm×mmatrixoverFandletY(x)bethematrix obtained from Y by replacing yij by xw(i,j)yij. Observe that to each row op- eration of the form adding a times row i to row j, for some j > i, performed on Y , there is a corresponding operation of adding xj−ia times row i to row j on Y (x) and this process preserves the degree function on positions; that is, if, for all (h, k), all terms in the (h, k) position of Y (x) have degree w(h, k) before the operation is performed, then this property also holds after the operation is performed.
Hence, corresponding to the factorization Pn,m = LQU,
we have a corresponding factorization
Pn,m(x) = H(x)Q(x)K(x),
where H(x), K(x) ∈ GL(m, F[x]) are lower and upper triangular matrices, re- spectively, and Q(x) has (i,j) entry xw(i,j)qij.