where N is a (n, n) square matrix and k ( 2) is a certain natural number. Then, N is
said to be a nilpotent matrix of an order k or a k-th order nilpotent matrix. If (12.61)
holds with k ¼ 1, N is a zero matrix.
Nilpotent matrices have following properties:
(i) Eigenvalues of a nilpotent matrix are zero. Let N be a k-th order nilpotent matrix.
Suppose that
Nx ¼ αx,
ð12:62Þ
where α is an eigenvalue and x (6 ¼0) is its corresponding eigenvector. Operating
N (k À 1) more times from the left of both sides of (12.62), we have
N
k x ¼ α
k x:
ð12:63Þ
Meanwhile, N
k
¼ 0 by definition, and so α
k
¼ 0, namely α ¼ 0.
Conversely, suppose that eigenvalues of a (n, n) matrix N are zero. From
Theorem 12.1, via a suitable similarity transformation with P we have
P
À1 NP ¼ e
N,
where e
N is a triangle matrix. Then, using (12.12) and (12.16) we have
f P
À1 NP x
ð Þ ¼ f e N
x
ð Þ ¼ f N x
ð Þ ¼ x
n
:
From Theorem 12.3, we have
f N N
ð Þ ¼ N
n
¼ 0:
Namely, N is a nilpotent matrix. In a trivial case, we have N ¼ 0
(zero matrix).
By Definition 12.2, we have N
k
¼ 0 with a k-th nilpotent (n, n) matrix N.
Then, its minimal polynomial is φ N (x) ¼ x
k (k n).
(ii) A nilpotent matrix N is not diagonalizable (except for a zero matrix). Suppose
that N is diagonalizable. Then N can be diagonalized by a non-singular matrix
P such that
P
À1 NP ¼ 0:
The above equation holds, because N only takes eigenvalues of zero. Operating
P from the left of the above equation and P
À1 from the right, we have
N ¼ 0:
12.3 Generalized Eigenvectors and Nilpotent Matrices
475