Linear Algebra, Theory And Applications, 2012a

258 LINEAR TRANSFORMATIONS CANONICAL FORMS
Proposition 10.5.1 Let the minimal polynomial of A ∈L(V,V ) be given by
p (λ) =
Then the eigenvalues of A are {λ 1 , ··· ,λ r }.
It follows from Corollary 10.2.4 that
r∏
(λ − λ k ) m k
k=1
V = ker(A − λ 1 I) m1 ⊕···⊕ker (A − λ r I) mr
≡
V 1 ⊕···⊕V r
where I denotes the identity linear transformation. Without loss of generality, let the
dimensions of the V k be decreasing from left to right. These V k are called the generalized
eigenspaces.
It follows from the definition of V k that (A − λ k I) is nilpotent on V k and clearly each
V k is A invariant. Therefore from Proposition 10.4.4, and letting A k denote the restriction
of A to V k , there exists an ordered basis for V k ,β k such that with respect to this basis, the
matrix of (A k − λ k I) is of the form given in that proposition, denoted here by J k .Whatis
the matrix of A k with respect to β k ? Letting {b 1 , ··· ,b r } = β k ,
A k b j =(A k − λ k I) b j + λ k Ib j ≡ ∑ s
J k sjb s + ∑ s
λ k δ sj b s = ∑ s
(
J
k
sj + λ k δ sj
)
bs
and so the matrix of A k with respect to this basis is
J k + λ k I
where I is the identity matrix. Therefore, with respect to the ordered basis {β 1 , ··· ,β r }
the matrix of A is in Jordan canonical form. This means the matrix is of the form
⎛
⎞
J (λ 1 ) 0
⎜
⎝
. ..
⎟
⎠ (10.5)
0 J (λ r )
where J (λ k )isanm k × m k matrix of the form
⎛
J k1 (λ k ) 0
J k2 (λ k )
⎜
⎝
. ..
0 J kr (λ k )
⎞
⎟
⎠
(10.6)
where k 1 ≥ k 2 ≥···≥k r ≥ 1and ∑ r
i=1 k i = m k . Here J k (λ) isak × k Jordan block of the
form
⎛
⎞
λ 1 0
. 0 λ ..
⎜
⎝
. .. . ⎟
(10.7)
..
1 ⎠
0 0 λ
This proves the existence part of the following fundamental theorem.
Note that if any of the β k consists of eigenvectors, then the corresponding Jordan block
will consist of a diagonal matrix having λ k down the main diagonal. This corresponds to