Page 64 - Textos de Matemática Vol. 44

54 R. GOW AND T. J. LAFFEY
and, using equivalence over Z[x], we add xJ−1 − I times the second block- mm
column of R to the first one to get
0 −Jm
R1:=J−1(xI−J)2 xI−J . mmmmm
Next, adding xJ−1 −I times the first block-row of R to the second, we obtain mm1
the matrix
and, therefore, to
This establishes the result when n = 2 and the general result follows by applying the same method systematically to the columns and rows of the Kronecker form of xImn −Jm ⊗Jn.
Hence the Smith invariants of xImn − Jm ⊗ Jn over F consist of the Smith invariants of (xIn −Jn)m or those of (xIm −Jm)n, together with the appropriate number of 1s, in order to bring the total number of invariants to mn.
Given positive integers m and n and an m × m matrix M, associate with each position (i, j) in an m × m matrix the weight
w(i, j) := i + j + n − m − 1.
(So, for example, the (1, m) position has weight n and the (m, m) position has weight n+m−1.)
We now prove:
Theorem 1.1. Let F be a field, and n ≥ m be positive integers. Let Pn,m be the m × m matrix
 n n ··· n n n 
m−1 m−2 2 1 0  n n ··· n n 0 m−2 m−3 1 0 
...... . ......,
 n n ··· 0 0 0  10
 n 0 ··· 0 0 0 0
interpreted as a matrix over F.
0 −Jm R2:= J−1(xI −J )2 0
,
and, on permuting columns, we deduce that R1 is equivalent over Z[x] to
mmm
−Jm 0 0 J−1(xI −J )2
mmm Im 0
0 (xIm−Jm)2 .