PUMaC 2009 · 加试 · 第 6 题

Problem 6.2 (8pts) . Calculate the signature of the lattice generated by: { (0 , 2 , 5 , 3) , (5 , 4 , 5 , 7) , (5 , 9 , 7 , 1) , (5 , 7 , 5 , 7) } (6.3) The signature is (1; 1; 3; 180). In general, a lattice can be represented by a matrix M with columns generating the lattice. The same lattice is generated if one replaces the columns by invertible linear combinations. One can also obtain an isomorphic lattice by taking bijective linear transformations of the n underlying Z which amounts to replacing the rows by invertible linear combinations. The goal is to compute the signature of the lattice by transforming the matrix M into a diagonal matrix. In this case M is the 4 by 4 matrix   0 5 5 5   2 4 9 7   .   5 5 7 5 3 7 1 7 6 By using the third column to eliminate the bottom row and one obtains   − 15 − 30 5 − 30   − 25 − 59 9 − 56   .   − 16 − 44 7 − 44 0 0 1 0 Using the fourth row, one can now elminate the entries in the third column and we are now left with the remaining three by three matrix   15 30 30   25 59 56 . 16 44 44 Subtracting the first row from the second and third, then clearing out the first column with the last row yields   0 − 180 − 180   0 − 111 − 114 . 1 14 14 Clearing out the bottom row now leaves us with a two by two matrix which we easily reduce in a few steps: ( ) ( ) ( ) 180 180 180 0 3 0 → → 111 114 111 3 0 180 This solution is very close to a proof of Theorem 6.1. See if you can figure it out. 7