Page 117 - Matrices theory and applications

6. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction
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extension, the pair (q, r) in the definition is uniquely defined by a = bq + r
and N(r) <N(b).
Proposition 6.1.3 Euclidean domains are principal ideal domains.
Proof
Let I be an ideal of a Euclidean domain A.If I = (0), there is nothing
to show. Otherwise, let us select in I\ {0} an element a of minimal norm.
If b ∈I, the remainder r of the Euclidean division of b by a is an element
of I and satisfies N(r) <N(a). The minimality of N(a) implies r =0, that
is, a|b. Finally, I =(a).
The converse of Proposition 6.1.3 is not true. For example, the quadratic
√
ring ZZ[ 14] is Euclidean, though not a principal ideal domain. More infor-
mation about rings of quadratic integers can be found in Cohn’s monograph
[10].
6.1.3 Elementary Matrices
An elementary matrix of order n is a matrix of one of the following forms:
• The transposition matrices: If σ ∈ S n ,the matrix P σ has entries
j
p ij = δ ,where δ is the Kronecker symbol.
σ(i)
• The matrices I n + aJ ik ,for a ∈ A and 1 ≤ i = k ≤ n,with

l m
(J ik ) lm = δ δ .
i k
• The diagonal invertible matrices, that is, those whose diagonal entries
areinvertiblein A.
We observe that the inverse of an elementary matrix is again elementary.
For example, (I n + aJ ik )(I n − aJ ik )= I n .
Theorem 6.1.1 A square invertible matrix of size n with entries in a
Euclidean domain A is a product of elementary matrices with entries in
A.

Proof
We shall prove the theorem for n = 2. The general case will be deduced
from that particular one and from the proof of Theorem 6.2.1 below, since
the matrices used in that proof are block-diagonal with 1 × 1and 2 × 2
diagonal blocks.
Let

a a 1
M =
c d