Page 118 - Matrices theory and applications

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be given in SL 2 (A): we have ad − a 1 c ∈ A .If N(a) <N(a 1 ), we multiply
M on the right by

1
.
1
0
We are now in the case N(a 1 ) ≤ N(a). Let a = a 1 q + a 2 be the Euclidean
division of a by a 1 .Then 0 6.2. Invariant Factors of a Matrix 101

1 0 a 2 a 1

M =: M = .
−q 1 · d
Next, we have

0 1 a 1 a 2
M =: M 1 = ,
1 0 · ·
with N(a 2 ) <N(a 1 ). We thus construct a sequence of matrices M k of the
form

a k−1 a k
,
· ·
with a k−1 = 0, each one the product of the previous one by elementary
matrices. Furthermore, N(a k ) <N(a k−1 ). From Proposition 6.1.2, this
sequence is finite, and there is a step for which a k =0. The matrix M k ,
being triangular and invertible, has an invertible diagonal D.Then M k D −1
has the form

1 0
,
· 1
whichis anelementarymatrix.
Again, the statement is false in a general principal ideal domain. Whether
GL n (A) equals the group spanned by elementary matrices is a difficult
question of Ktheory.

6.2 Invariant Factors of a Matrix

Theorem 6.2.1 Let M ∈ M n×m (A) be amatrixwithentries in aprincipal
ideal domain. Then there exist two invertible matrices P ∈ GL n (A), Q ∈
GL m (A) and a quasi-diagonal matrix D ∈ M n×m (A) (that is, d ij =0 for
i = j) such that:

• on the one hand, M = PDQ,
• on the other hand, d 1 |d 2 ,... ,d i |d i+1 ,... ,where the d j are the
diagonal entries of D.