Page 244 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Forget about type (c) operations for the moment, and concentrate on types (a) and (b), which
change only row r. If r is not one of the indices i , ... , i , then
1 k
D , (e(M)) = D , (M).
I J I J
If r is among the indices i , ..., i , then in the two cases we have
1 k
(a) D , (e(M)) = D ( , ..., c , ..., )
I J J i1 r ik
= cD ( , ..., , ..., )
J i1 r ik
= cD (M);
I, J
(b) D (e(M)) = D ( , ..., + g , ..., )
I, J J i1 r s ik
= D (M) +gD ( , ..., , ..., )
I, J J i1 s ik
For type (a) operations, it is clear that any f which divides D (M) also divides D , (e(M)). For the
I,J I J
case of a type (c) operation, notice that
D ( , ..., , ..., ) = 0, if s = i, for some j
J i1 s ik
D ( , ..., ..., ) = ± D ’ (M), if s i, for all j.
J i1 s ik I . J
The I’ in the last equation is the k-tuple (i , ... , s, ... , i ) arranged in increasing order. It should now
1 k
be apparent that, if f divides every D (M), then f divides every D (e(M)).
I.J I.J
Operations of type (c) can be taken care of by roughly the same argument or by using the fact
that such an operation can be effected by a sequence of operations of types (a) and (b).
Corollary: Each matrix M in F[x] m n is equivalent to precisely one matrix N which is in normal
form. The polynomials f , ... , f which occur on the main diagonal of N are
1 k
(M )
k
f = , 1 k min (m, n)
k (M )
k 1
where, for convenience, we define (M) = l.
0
Proof: If N is in normal form with diagonal entries f , ..., f ; it is quite easy to see that
1 k
(N) = f f ... f .
k 1 2 k
Of course, we call the matrix N in the last corollary the normal form of M. The polynomials f , ...,
1
f are often called the invariant factors of M.
k
Suppose that A is an n n matrix with entries in F, and let p , ... , p be the invariant factors for A.
1 r
We now see that the normal form of the matrix xI – A has diagonal entries 1, 1, ... , 1, p , ..., p . The
r l
last corollary tells us what p , ... , p are, in terms of submatrices of xI – A. The number n – r is the
1 r
largest k such that (xI – A) = 1. The minimal polynomial p is the characteristic polynomial for
k 1
A divided by the greatest common divisor of the determinants of all (n – 1) (n – 1) submatrices
of xI – A, etc.
Self Assessment

1. True or false? Every matrix in F n × n is row-equivalent to an upper-triangular matrix.
2. T be a linear operator on a finite dimensional vector space and let A be the matrix of T in
some ordered basis. Show that T has a cyclic vector if and only if the determinants of the
(n – 1) (n – 1) sub-matrices of (xI – A) are relatively prime.

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