Page 125 - Matrices theory and applications

6. Matrices with Entries in a Principal Ideal Domain; Jordan Reduction
108
Theorem 6.3.5 Let k be a field, M ∈ M n (k) a square matrix, and
P 1 ,... ,P n its similarity invariants. Then P n is the minimal polynomial
of M. In particular, the minimal polynomial does not depend on the field
under consideration, as long as it contains the entries of M.
Proof

We use the first canonical form M of M.Since M and M are similar,

they have the same minimal polynomial. One thus can assume that M is
in the canonical form M =diag(M 1 ,... ,M n ), where M j is the companion
matrix of P j .Since P j (M j ) = 0 (Cayley–Hamilton, theorem 2.5.1) and
P j |P n ,wehave P n (M j ) = 0 and thus P n (M)= 0 n . Hence, the minimal
polynomial Q M divides P n .Conversely, Q(M)= 0 n implies Q(M n )=0.
Since P n is the minimal polynomial of M n , P n divides Q. Finally, P n = Q M .
Finally, since the similarity invariants do not depend on the choice of the
field, P n also does not depend on this choice.
Warning: One may draw an incorrect conclusion if one applies Theorem
6.3.5 carelessly. Given a matrix M ∈ M n (ZZ), one can define a matrix
M (p) in M n (ZZ/pZZ) by reduction modulo p (p aprime number). But the
minimal polynomial of M (p) is not necessarily the reduction modulo p of
Q M . Here is an example: Let us take n =2 and

2 2
M = .
0 2
2
Then Q M divides P M =(X − 2) , but Q M = X − 2, since M =2I 2 . Hence
2
Q M =(X −2) . On the other hand, M (2) =0 2 , whose minimal polynomial
2
is X, which is different from X , the reduction modulo 2 of Q M .
The explanation of this phenomenon is the following. The matrices M
and M (2) are composed of scalars of different natures. There is no field
L containing simultaneously ZZ and ZZ/2ZZ. There is thus no context in
which Theorem 6.3.5 could be applied.

6.3.3 Second Canonical Form of a Square Matrix
We now decompose the similarity invariants of M into products of irre-
ducible polynomials. This decomposition depends, of course, on the choice
of the field of scalars. Denoting by p 1 ,... ,p t the list of distinct irreducible
(in k[X]) factors of P n ,we have
t
α(j,k)
P j = p , 1 ≤ j ≤ n
k
k=1
(because P j divides P n ), where the α(j, k) are nondecreasing with respect
to j,since P j divides P j+1 .