Page 238 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes m n
We will be concerned with F ( ) , the collection of m n matrices with entries which are
x
polynomials over the field F. If M is such a matrix, an elementary row operation on M is one of
the following:
1. multiplications of one row of M by a non-zero scalar in F;
2. replacement of the rth row of M by row r plus f times row s, where f is any polynomial
over F and r = s;
3. interchange of two rows of M.
The inverse operation of an elementary row operation is an elementary row operation of the
same type. Notice that we could not make such an assertion if we allowed non-scalar polynomials
in (1). An m × m elementary matrix, that is, an elementary matrix in F[x] m m , is one which can be
obtained from the m m identity matrix by means of a single elementary row operation. Clearly
each elementary row operation on M can be effected by multiplying M on the left by a suitable
m m elementary matrix; in fact, if e is the operation, then
e(M) = e(I)M.
Let M, N be matrices in F[x] m n . We say that N is row-equivalent to M if N can be obtained from
M by a finite succession of elementary row operations:
M = M M ... M = N.
0 1 k
Evidently N is row-equivalent to M if and only if M is row-equivalent to N, so that we may use
the terminology 'M and N are row-equivalent.' If N is row-equivalent to M, then
N = PM
where the m m matrix P is a product of elementary matrices:

P = E ... E .
1 k
In particular, P is an invertible matrix with inverse
P = E ... E .
–1
–l
–l
k 1
Of course, the inverse of E, comes from the inverse elementary row operation.
All of this is just as it is in the case of matrices with entries in F. Thus, the next problem which
suggests itself is to introduce a row-reduced echelon form for polynomial matrices. Here, we
meet a new obstacle. How do we row-reduce a matrix? The first step is to single out the leading
non-zero entry of row 1 and to divide every entry of row 1 by that entry. We cannot (necessarily)
do that when the matrix has polynomial entries. As we shall see in the next theorem, we can
circumvent this difficulty in certain cases; however, there is not any entirely suitable
row-reduced form for the general matrix in F[x] m n . If we introduce column operations as well
and study the type of equivalence which results from allowing the use of both types of operations,
we can obtain a very useful standard form for each matrix. The basic tool is the following.
Lemma: Let M be a matrix in F[x] m n which has some non-zero entry in its first column, and let p
be the greatest common divisor of the entries in column 1 of M. Then M is row-equivalent to a
matrix N which has
p
0

0

as its first column.

232 LOVELY PROFESSIONAL UNIVERSITY