Linear Algebra, Theory And Applications, 2012a

10.3. CYCLIC SETS 251
while
⎛ ⎞
0
.
Aq
x
≡ A ∑
⎜ ⎟
⎝
j
. ⎠
0
x j v k j = ∑ j
x j A k v k j = ∑ j
x j
∑
i
M k ijv k i
because, as discussed earlier, Avj k = ∑ i M ij k vk i because M k is the matrix of A k with respect
to the basis β k .
An examination of the proof of the above theorem yields the following corollary.
Corollary 10.2.7 If any β k in the above consists of eigenvectors, then M k is a diagonal
matrix having the corresponding eigenvalues down the diagonal.
It follows that it would be interesting to consider special bases for the vector spaces in
the direct sum. This leads to the Jordan form or more generally other canonical forms such
as the rational canonical form.
10.3 Cyclic Sets
ItwasshownabovethatforA ∈L(V,V )forV a finite dimensional vector space over the
field of scalars F, there exists a direct sum decomposition
V = V 1 ⊕···⊕V q
where
V k =ker(φ k (A) m k
)
and φ k (λ) is an irreducible polynomial. Here the minimal polynomial of A was
q∏
φ k (λ) m k
k=1
Next I will consider the problem of finding a basis for V k such that the matrix of A
restricted to V k assumes various forms.
Definition 10.3.1 Letting x ≠0denote by β x the vectors { x, Ax, A 2 x, ··· ,A m−1 x } where
m is the smallest such that A m x ∈ span ( x, ··· ,A m−1 x ) . This is called an A cyclic set.
The vectors which result are also called a Krylov sequence.
The following is the main idea.
diagram.
To help organize the ideas in this lemma, here is a
ker(φ(A) m )
U ⊆ ker(φ(A))
W
v 1 , ..., v s
x 1 ,x 2 , ..., x p