
Chapter 8
Linear Algebra over
Polynomial Rings
8.1 Introduction
Over a field F, to determine whether two m×n matrices A and B belong to
the same orbit under the left action of GL
m
(F), we compute the row canonical
forms (RCFs, also called Gauss–Jordan forms, or reduced row echelon forms)
of A and B and check whether they are equal. Similarly, for the left-right
action of GL
m
(F) × GL
n
(F), we compute the Smith normal forms Smith(A)
and Smith(B) and check whether they are equal.
Over a Euclidean domain, in particular the ring F[x] of polynomials in
one variable x over the field F, a modification of Gaussian elimination gives a
similar result. Since the domain ...