Page 46 - Matrices theory and applications

−1

since M = PM P
and M =0 imply M = 0. Hence,
00
is not diagonalizable.
2.7 Trigonalization 01 2.7. Trigonalization 29
Let us begin with an application of the Cayley–Hamilton theorem.
Proposition 2.7.1 Let M ∈ M n (K) and let P M be its characteristic poly-
n
nomial. If P M = QR with coprime factors Q, R ∈ K[X],then K = E ⊕F,
where E, F are the ranges of Q(M) and R(M), respectively. Moreover, one
has E =ker R(M), F =ker Q(M).
More generally, if P M = R 1 ··· R s ,where the R s are coprime, one has
n
K = E 1 ⊕ ··· ⊕ E s with E j =ker R j (M).
Proof
It is sufficient to prove the first assertion. From B´ezout’s theorem, there
n
exists R 1 ,Q 1 ∈ K[X] such that RR 1 + QQ 1 = 1. Hence, every x ∈ K can
be written as a sum y + z with y = Q(M)(Q 1 (M)x) ∈ E, and similarly
n
z = R(M)(R 1 (M)x) ∈ F. Hence K = E + F.
Furthermore, for every y ∈ E, the Cayley–Hamilton theorem says that
R(M)y = 0. Likewise, z ∈ F implies Q(M)z =0. If x ∈ E ∩ F,one has
thus R(M)x = Q(M)x = 0. Again using B´ezout’s theorem, one obtains
n
x =0. This proves K = E ⊕ F.
Finally, E ⊂ ker R(M). Since these two vector spaces have the same
dimension (namely n − dim F), they are equal.

If K is algebraically closed, we can split P M in the form

P M (X)= (X − λ) n λ .
λ∈Sp(M)
n
From Proposition 2.7.1 one has K = ⊕ λ E λ ,where E λ =ker(M − λI) n λ
is called a generalized eigenspace. Choosing a basis in each E λ ,we obtain a
n
new basis B of K .If P is the matrix of the linear transformation from the
canonical basis to B,the matrix PMP −1 is block-diagonal, because each
E λ is stable under the action of M:
PMP −1 =diag(... ,M λ ,... ).

The matrix M λ is that of the restriction of M to E λ .Since E λ =ker(M −
λI) n λ , one has (M λ − λI) n λ =0, so that λ is the unique eigenvalue of M λ .