Page 243 - DMTH502_LINEAR_ALGEBRA

Unit 22: Computation of Invariant Factors

(t)
we must arrive at a matrix M which has the form (9): because at each successive step we have Notes
l(M (k + 1) < l(M . We name the process which we have just defined P7-36.
(k)
t
M (0) PL-36 M ( )
In (9), the polynomial g may or may not divide every entry of S. If it does not, find the first
column which has an entry not divisible by g and add that column to column 1. The new first
column contains both g and an entry gh + r where r 0 and deg r < deg g. Apply process P7-36 and
the result will be another matrix of the form (9), where the degree of the corresponding g has
decreased.
It should now be obvious that in a finite number of steps we will obtain (8), i.e., we will reach a
matrix of the form (9) where the degree of g cannot be further reduced.
We want to show that the normal form associated with a matrix M is unique. Two things we
have seen provide clues as to how the polynomials f ..., f in Theorem 4 are uniquely determined
l 1
by M. First, elementary row and column operations do not change the determinant of a square
matrix by more than a non-zero scalar factor. Second, elementary row and column operations
do not change the greatest common divisor of the entries of a matrix.
Definition: Let M be an m n matrix with entries in F[x]. If 1 k min (m, n), we define (M) to
k
be the greatest common divisor of the determinants of all k k submatrices of M.
Recall that a k k submatrix of M is one obtained by deleting some m – k rows and some n – k
columns of M. In other words, we select certain k-tuples
I = (i , ..., i ), 1 i < ... < i m
1 k 1 k
J = (j , ..., j ), 1 j , < ... < j n
1 k 1 k
and look at the matrix formed using those rows and columns of M. We are interested in the
determinants

M i 1 1 j  M i 1 k j
D , (M) = det   ...(12)
I J
M  M i
k i 1 j k k j
The polynomial (M) is the greatest common divisor of the polynomials D , (M), as I and J range
k I j
over the possible k-tuples.
Theorem 5: If M and N are equivalent m n matrices with entries in F[x], then
(M) = (N), 1 k min (m, n) ...(13)
k k
Proof: It will suffice to show that a single elementary row operation e does not change . Since
k
the inverse of e is also an elementary row operation, it will suffice to show this: If a polynomial
f divides every D , (M) , then f divides D , (e(M)) for all k-tuples I and J.
I J I J
Since we are considering a row operation, let , ..., be the rows of M and let us employ the
1 m
notation
D ( ..., ) = D , (M).
J i1 ik I J
Given I and J, what is the relation between D , (M) and D , (e(M))? Consider the three types of
I J I J
operations e:
(a) multiplication of row r by a non-zero scalar c;
(b) replacement of row r by row r plus g times row s, r s;

(c) interchange of rows r and s, r s.

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