properties of the matrices. In this section we examine basic properties of diagonalizable matrices.
In Sect 12.6.1 we have shown that Span{N
i À 1 a ρ , N
i À 2 a ρ , Á Á ÁNa ρ , a ρ } forms a
N-invariant subspace of a dimension i, where a ρ satisfies the relations N
i a ρ ¼ 0 and
N
i À 1 a ρ 6 ¼ 0 (r i + 1 + 1 j r i ) as in (12.122). Of these vectors, only N
i À 1 a ρ is a sole
proper eigenvector that is accompanied by (i À 1) generalized eigenvectors. Note
that only the proper eigenvector can construct one-dimensional N-invariant subspace
by itself. This is because regarding other generalized eigenvectors g (here g stands
for all of generalized eigenvectors), Ng (≠0) and g are linearly independent. Note
that with a proper eigenvector e, we have Ne = 0. A corresponding Jordan block is
represented by a matrix as given in (12.122) in reference to the basis vectors
comprising these i eigenvectors. Therefore, if a (n, n) matrix A has only proper
eigenvectors, all Jordan blocks are one-dimensional. This means that A is diagonalizable. That A has only proper eigenvectors is equivalent to that those eigenvectors
form a one-dimensional subspace and that V
n is a direct sum of the subspaces
spanned by individual proper eigenvectors. In other words, if V
n is a direct sum of
subspaces (i.e., eigenspaces) spanned by individual proper eigenvectors of A, A is
diagonalizable.
Next, suppose that A is diagonalizable. Then, after an appropriate similarity
transformation with a non-singular matrix P, A has a following form:
P
À1 AP ¼
α 1
⋱
α 1
α 2
⋱
α 2
⋱
α s
⋱
α s
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
ð12:181Þ
In this case, let us examine what form a minimal polynomial φ A (x) for A takes. A
characteristic polynomial f A (x) for A is invariant through similarity transformation,
so is φ A (x). That is,
φ P
À1 AP x
ð Þ ¼ φ A x
ð Þ:
ð12:182Þ
From (12.181), we find that A À α i E (1 i s) has a “perforated” form such as
(12.143) with the diagonalized form unchanged. Then we have
12.7 Diagonalizable Matrices
513
In Sect 12.6.1 we have shown that Span{N
i À 1 a ρ , N
i À 2 a ρ , Á Á ÁNa ρ , a ρ } forms a
N-invariant subspace of a dimension i, where a ρ satisfies the relations N
i a ρ ¼ 0 and
N
i À 1 a ρ 6 ¼ 0 (r i + 1 + 1 j r i ) as in (12.122). Of these vectors, only N
i À 1 a ρ is a sole
proper eigenvector that is accompanied by (i À 1) generalized eigenvectors. Note
that only the proper eigenvector can construct one-dimensional N-invariant subspace
by itself. This is because regarding other generalized eigenvectors g (here g stands
for all of generalized eigenvectors), Ng (≠0) and g are linearly independent. Note
that with a proper eigenvector e, we have Ne = 0. A corresponding Jordan block is
represented by a matrix as given in (12.122) in reference to the basis vectors
comprising these i eigenvectors. Therefore, if a (n, n) matrix A has only proper
eigenvectors, all Jordan blocks are one-dimensional. This means that A is diagonalizable. That A has only proper eigenvectors is equivalent to that those eigenvectors
form a one-dimensional subspace and that V
n is a direct sum of the subspaces
spanned by individual proper eigenvectors. In other words, if V
n is a direct sum of
subspaces (i.e., eigenspaces) spanned by individual proper eigenvectors of A, A is
diagonalizable.
Next, suppose that A is diagonalizable. Then, after an appropriate similarity
transformation with a non-singular matrix P, A has a following form:
P
À1 AP ¼
α 1
⋱
α 1
α 2
⋱
α 2
⋱
α s
⋱
α s
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
ð12:181Þ
In this case, let us examine what form a minimal polynomial φ A (x) for A takes. A
characteristic polynomial f A (x) for A is invariant through similarity transformation,
so is φ A (x). That is,
φ P
À1 AP x
ð Þ ¼ φ A x
ð Þ:
ð12:182Þ
From (12.181), we find that A À α i E (1 i s) has a “perforated” form such as
(12.143) with the diagonalized form unchanged. Then we have
12.7 Diagonalizable Matrices
513