c 1 N
kÀ1 x ¼ 0,
ð12:68Þ
implying that c 1 ¼ 0. Continuing this procedure, we finally get c i ¼ 0 (0 i k À 1).
This completes the proof.
This immediately tells us that for a k-th order nilpotent matrix N : V
n
! V
n , we
should have k n. This is because the number of linearly independent vectors is at
most n.
In Example 12.4, N causes a transformation of basis vectors in V
2 such that
e 1 e 2
ð
ÞN ¼ e 1 e 2
ð
Þ
0 1
0 0
¼ 0 e 1
ð
Þ:
That is, Ne 2 ¼ e 1 . Then, linearly independent two vectors e 2 and Ne 2 correspond
to the case of Theorem 12.4.
So far we have shown simple cases where matrices can be diagonalized via
similarity transformation. This is equivalent to that the relevant vector space is
decomposed to a direct sum of (invariant) subspaces. Nonetheless, if a characteristic
polynomial of the matrix has multiple root(s), it remains uncertain whether the vector
space is decomposed to such a direct sum. To answer this question, we need a
following lemma.
∎
Lemma 12.1 Let f 1 (x), f 2 (x), Á Á Á, and f s (x) be polynomials without a common factor.
Then we have s polynomials M 1 (x), M 2 (x), Á Á Á, M s (x) that satisfy the following
relation:
M 1 x
ð Þf 1 x
ð Þ þ M 2 x
ð Þf 2 x
ð Þ þ Á Á Á þ M s x
ð Þf s x
ð Þ ¼ 1:
ð12:69Þ
Proof Let M i (x) (1 i s) be arbitrarily chosen polynomials and deal with a set of
g(x) that can be expressed as
g x
ð Þ ¼ M 1 x
ð Þf 1 x
ð Þ þ M 2 x
ð Þf 2 x
ð Þ þ Á Á Á þ M s x
ð Þf s x
ð Þ:
ð12:70Þ
Let a whole set of g(x) be H. Then H has following two properties:
(i) g 1 x
ð Þ, g 2 x
ð Þ 2 H⟹g 1 x
ð Þ þ g 2 x
ð Þ 2 H,
(ii) g x
ð Þ 2 H, M(x): an arbitrarily chosen polynomial ⟹M x
ð Þg x
ð Þ 2 H.
Now let the lowest order polynomial out of the set H be g 0 (x). Then 8g x
ð Þ 2 H
ð
Þ
are a multiple of g 0 (x). Suppose that dividing g(x) by g 0 (x), we have
g x
ð Þ ¼ M x
ð Þg 0 x
ð Þ þ h x
ð Þ,
ð12:71Þ
where h(x) is a certain polynomial. Since g x
ð Þ, g 0 x
ð Þ 2 H, we have h x
ð Þ 2 H from
the above properties (i) and (ii). If h(x) 6 ¼ 0, the order of h(x) is lower than that of
12.3 Generalized Eigenvectors and Nilpotent Matrices
477
kÀ1 x ¼ 0,
ð12:68Þ
implying that c 1 ¼ 0. Continuing this procedure, we finally get c i ¼ 0 (0 i k À 1).
This completes the proof.
This immediately tells us that for a k-th order nilpotent matrix N : V
n
! V
n , we
should have k n. This is because the number of linearly independent vectors is at
most n.
In Example 12.4, N causes a transformation of basis vectors in V
2 such that
e 1 e 2
ð
ÞN ¼ e 1 e 2
ð
Þ
0 1
0 0
¼ 0 e 1
ð
Þ:
That is, Ne 2 ¼ e 1 . Then, linearly independent two vectors e 2 and Ne 2 correspond
to the case of Theorem 12.4.
So far we have shown simple cases where matrices can be diagonalized via
similarity transformation. This is equivalent to that the relevant vector space is
decomposed to a direct sum of (invariant) subspaces. Nonetheless, if a characteristic
polynomial of the matrix has multiple root(s), it remains uncertain whether the vector
space is decomposed to such a direct sum. To answer this question, we need a
following lemma.
∎
Lemma 12.1 Let f 1 (x), f 2 (x), Á Á Á, and f s (x) be polynomials without a common factor.
Then we have s polynomials M 1 (x), M 2 (x), Á Á Á, M s (x) that satisfy the following
relation:
M 1 x
ð Þf 1 x
ð Þ þ M 2 x
ð Þf 2 x
ð Þ þ Á Á Á þ M s x
ð Þf s x
ð Þ ¼ 1:
ð12:69Þ
Proof Let M i (x) (1 i s) be arbitrarily chosen polynomials and deal with a set of
g(x) that can be expressed as
g x
ð Þ ¼ M 1 x
ð Þf 1 x
ð Þ þ M 2 x
ð Þf 2 x
ð Þ þ Á Á Á þ M s x
ð Þf s x
ð Þ:
ð12:70Þ
Let a whole set of g(x) be H. Then H has following two properties:
(i) g 1 x
ð Þ, g 2 x
ð Þ 2 H⟹g 1 x
ð Þ þ g 2 x
ð Þ 2 H,
(ii) g x
ð Þ 2 H, M(x): an arbitrarily chosen polynomial ⟹M x
ð Þg x
ð Þ 2 H.
Now let the lowest order polynomial out of the set H be g 0 (x). Then 8g x
ð Þ 2 H
ð
Þ
are a multiple of g 0 (x). Suppose that dividing g(x) by g 0 (x), we have
g x
ð Þ ¼ M x
ð Þg 0 x
ð Þ þ h x
ð Þ,
ð12:71Þ
where h(x) is a certain polynomial. Since g x
ð Þ, g 0 x
ð Þ 2 H, we have h x
ð Þ 2 H from
the above properties (i) and (ii). If h(x) 6 ¼ 0, the order of h(x) is lower than that of
12.3 Generalized Eigenvectors and Nilpotent Matrices
477