Page 121 - Matrices theory and applications

6. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction
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factor may occur several times in the list d 1 ,... ,d r ,upto association. The
number of times that a factor d or its associates occur is its multiplicity.
If m = n and if the invariant factors of a matrix M are (1, ··· , 1), then
D = I n ,and M = PQ is invertible. Conversely, if M is invertible, then the
decomposition M = MI n I n shows that d 1 = ··· = d n =1.
If A is a field, then there are only two ideals: A = (1) itself and (0). The
list of invariant factors of a matrix is thus of the form (1,... , 1, 0,... , 0).
Of course, there may be no 1’s (for the matrix 0 m×n), or no 0’s. There are
thus exactly min(n, m) + 1 classes of equivalent matrices in M n (A), two
matrices being equivalent if and only if they have the same rank q.The rank
is then the number of 1’s among the invariant factors. The decomposition
M = PDQ is then called the rank decomposition.
Theorem 6.2.2 Let k be afieldand M ∈ M n×m(k) amatrix. Let q be
n
the rank of M, that is, the dimension of the linear subspace of k spanned
by the columns of M. Then there exist two square invertible matrices P, Q
such that M = PDQ with d ii =1 if i ≤ q and d ij =0 in all other cases.
6.3 Similarity Invariants and Jordan Reduction

From now on, k will denote a field and A = k[X] the ring of polynomi-
als over k. This ring is Euclidean, hence a principal ideal domain. In the
sequel, the results are effective, in the sense that the normal forms that
we define will be obtained by means of an algorithm that uses right or left
multiplications by elementary matrices of M n (A), the computations being
based upon the Euclidean division of polynomials.
Given a matrix B ∈ M n (k) (a square matrix with constant entries, in the
sense that they are not polynomials), we consider the matrix XI n − B ∈
M n (A), where X is the indeterminate in A.
Definition 6.3.1 If B ∈ M n (k), the invariant factors of M := XI n − B
are called invariant polynomials of B,or similarity invariants of B.

This definition is justified by the following statement.
Theorem 6.3.1 Two matrices in M n (k) are similar if and only if
they have the same list of invariant polynomials (counted with their
multiplicities).

This theorem is a particular case of a more general one:
Theorem 6.3.2 Let A 0 ,A 1 ,B 0 ,B 1 be matrices in M n (k),with A 0 ,A 1 .
Then the matrices XA 0 + B 0 and XA 1 + B 1 are equivalent (in M n (A))if
and only if there exist G, H ∈ GL n (k) such that
GA 0 = A 1 H, GB 0 = B 1 H.