Mathematical concept in algebra
In
linear algebra , a nilpotent matrix is a
square matrix N such that
N
k
=
0
{\displaystyle N^{k}=0\,}
for some positive
integer
k
{\displaystyle k}
. The smallest such
k
{\displaystyle k}
is called the index of
N
{\displaystyle N}
,
[1] sometimes the degree of
N
{\displaystyle N}
.
More generally, a nilpotent transformation is a
linear transformation
L
{\displaystyle L}
of a
vector space such that
L
k
=
0
{\displaystyle L^{k}=0}
for some positive integer
k
{\displaystyle k}
(and thus,
L
j
=
0
{\displaystyle L^{j}=0}
for all
j
≥
k
{\displaystyle j\geq k}
).
[2]
[3]
[4] Both of these concepts are special cases of a more general concept of
nilpotence that applies to elements of
rings .
Examples
Example 1
The matrix
A
=
0
1
0
0
{\displaystyle A={\begin{bmatrix}0&1\\0&0\end{bmatrix}}}
is nilpotent with index 2, since
A
2
=
0
{\displaystyle A^{2}=0}
.
Example 2
More generally, any
n
{\displaystyle n}
-dimensional
triangular matrix with zeros along the
main diagonal is nilpotent, with index
≤
n
{\displaystyle \leq n}
[
citation needed ] . For example, the matrix
B
=
0
2
1
6
0
0
1
2
0
0
0
3
0
0
0
0
{\displaystyle B={\begin{bmatrix}0&2&1&6\\0&0&1&2\\0&0&0&3\\0&0&0&0\end{bmatrix}}}
is nilpotent, with
B
2
=
0
0
2
7
0
0
0
3
0
0
0
0
0
0
0
0
;
B
3
=
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
{\displaystyle B^{2}={\begin{bmatrix}0&0&2&7\\0&0&0&3\\0&0&0&0\\0&0&0&0\end{bmatrix}};\ B^{3}={\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}}}
The index of
B
{\displaystyle B}
is therefore 3.
Example 3
Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,
C
=
5
−
3
2
15
−
9
6
10
−
6
4
C
2
=
0
0
0
0
0
0
0
0
0
{\displaystyle C={\begin{bmatrix}5&-3&2\\15&-9&6\\10&-6&4\end{bmatrix}}\qquad C^{2}={\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}}}
although the matrix has no zero entries.
Example 4
Additionally, any matrices of the form
a
1
a
1
⋯
a
1
a
2
a
2
⋯
a
2
⋮
⋮
⋱
⋮
−
a
1
−
a
2
−
…
−
a
n
−
1
−
a
1
−
a
2
−
…
−
a
n
−
1
…
−
a
1
−
a
2
−
…
−
a
n
−
1
{\displaystyle {\begin{bmatrix}a_{1}&a_{1}&\cdots &a_{1}\\a_{2}&a_{2}&\cdots &a_{2}\\\vdots &\vdots &\ddots &\vdots \\-a_{1}-a_{2}-\ldots -a_{n-1}&-a_{1}-a_{2}-\ldots -a_{n-1}&\ldots &-a_{1}-a_{2}-\ldots -a_{n-1}\end{bmatrix}}}
such as
5
5
5
6
6
6
−
11
−
11
−
11
{\displaystyle {\begin{bmatrix}5&5&5\\6&6&6\\-11&-11&-11\end{bmatrix}}}
or
1
1
1
1
2
2
2
2
4
4
4
4
−
7
−
7
−
7
−
7
{\displaystyle {\begin{bmatrix}1&1&1&1\\2&2&2&2\\4&4&4&4\\-7&-7&-7&-7\end{bmatrix}}}
square to zero.
Example 5
Perhaps some of the most striking examples of nilpotent matrices are
n
×
n
{\displaystyle n\times n}
square matrices of the form:
2
2
2
⋯
1
−
n
n
+
2
1
1
⋯
−
n
1
n
+
2
1
⋯
−
n
1
1
n
+
2
⋯
−
n
⋮
⋮
⋮
⋱
⋮
{\displaystyle {\begin{bmatrix}2&2&2&\cdots &1-n\\n+2&1&1&\cdots &-n\\1&n+2&1&\cdots &-n\\1&1&n+2&\cdots &-n\\\vdots &\vdots &\vdots &\ddots &\vdots \end{bmatrix}}}
The first few of which are:
2
−
1
4
−
2
2
2
−
2
5
1
−
3
1
5
−
3
2
2
2
−
3
6
1
1
−
4
1
6
1
−
4
1
1
6
−
4
2
2
2
2
−
4
7
1
1
1
−
5
1
7
1
1
−
5
1
1
7
1
−
5
1
1
1
7
−
5
…
{\displaystyle {\begin{bmatrix}2&-1\\4&-2\end{bmatrix}}\qquad {\begin{bmatrix}2&2&-2\\5&1&-3\\1&5&-3\end{bmatrix}}\qquad {\begin{bmatrix}2&2&2&-3\\6&1&1&-4\\1&6&1&-4\\1&1&6&-4\end{bmatrix}}\qquad {\begin{bmatrix}2&2&2&2&-4\\7&1&1&1&-5\\1&7&1&1&-5\\1&1&7&1&-5\\1&1&1&7&-5\end{bmatrix}}\qquad \ldots }
These matrices are nilpotent but there are no zero entries in any powers of them less than the index.
[5]
Example 6
Consider the linear space of
polynomials of a bounded degree. The
derivative operator is a linear map. We know that applying the derivative to a polynomial decreases its degree by one, so when applying it iteratively, we will eventually obtain zero. Therefore, on such a space, the derivative is representable by a nilpotent matrix.
Characterization
For an
n
×
n
{\displaystyle n\times n}
square matrix
N
{\displaystyle N}
with
real (or
complex ) entries, the following are equivalent:
N
{\displaystyle N}
is nilpotent.
The
characteristic polynomial for
N
{\displaystyle N}
is
det
(
x
I
−
N
)
=
x
n
{\displaystyle \det \left(xI-N\right)=x^{n}}
.
The
minimal polynomial for
N
{\displaystyle N}
is
x
k
{\displaystyle x^{k}}
for some positive integer
k
≤
n
{\displaystyle k\leq n}
.
The only complex eigenvalue for
N
{\displaystyle N}
is 0.
The last theorem holds true for matrices over any
field of characteristic 0 or sufficiently large characteristic. (cf.
Newton's identities )
This theorem has several consequences, including:
The index of an
n
×
n
{\displaystyle n\times n}
nilpotent matrix is always less than or equal to
n
{\displaystyle n}
. For example, every
2
×
2
{\displaystyle 2\times 2}
nilpotent matrix squares to zero.
The
determinant and
trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be
invertible .
The only nilpotent
diagonalizable matrix is the zero matrix.
See also:
Jordan–Chevalley decomposition#Nilpotency criterion .
Classification
Consider the
n
×
n
{\displaystyle n\times n}
(upper)
shift matrix :
S
=
0
1
0
…
0
0
0
1
…
0
⋮
⋮
⋮
⋱
⋮
0
0
0
…
1
0
0
0
…
0
.
{\displaystyle S={\begin{bmatrix}0&1&0&\ldots &0\\0&0&1&\ldots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\ldots &1\\0&0&0&\ldots &0\end{bmatrix}}.}
This matrix has 1s along the
superdiagonal and 0s everywhere else. As a linear transformation, the shift matrix "shifts" the components of a vector one position to the left, with a zero appearing in the last position:
S
(
x
1
,
x
2
,
…
,
x
n
)
=
(
x
2
,
…
,
x
n
,
0
)
.
{\displaystyle S(x_{1},x_{2},\ldots ,x_{n})=(x_{2},\ldots ,x_{n},0).}
[6]
This matrix is nilpotent with degree
n
{\displaystyle n}
, and is the
canonical nilpotent matrix.
Specifically, if
N
{\displaystyle N}
is any nilpotent matrix, then
N
{\displaystyle N}
is
similar to a
block diagonal matrix of the form
S
1
0
…
0
0
S
2
…
0
⋮
⋮
⋱
⋮
0
0
…
S
r
{\displaystyle {\begin{bmatrix}S_{1}&0&\ldots &0\\0&S_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &S_{r}\end{bmatrix}}}
where each of the blocks
S
1
,
S
2
,
…
,
S
r
{\displaystyle S_{1},S_{2},\ldots ,S_{r}}
is a shift matrix (possibly of different sizes). This form is a special case of the
Jordan canonical form for matrices.
[7]
For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix
0
1
0
0
.
{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}.}
That is, if
N
{\displaystyle N}
is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b 1 , b 2 such that N b 1 = 0 and N b 2 = b 1 .
This
classification theorem holds for matrices over any
field . (It is not necessary for the field to be algebraically closed.)
Flag of subspaces
A nilpotent transformation
L
{\displaystyle L}
on
R
n
{\displaystyle \mathbb {R} ^{n}}
naturally determines a
flag of subspaces
{
0
}
⊂
ker
L
⊂
ker
L
2
⊂
…
⊂
ker
L
q
−
1
⊂
ker
L
q
=
R
n
{\displaystyle \{0\}\subset \ker L\subset \ker L^{2}\subset \ldots \subset \ker L^{q-1}\subset \ker L^{q}=\mathbb {R} ^{n}}
and a signature
0
=
n
0
<
n
1
<
n
2
<
…
<
n
q
−
1
<
n
q
=
n
,
n
i
=
dim
ker
L
i
.
{\displaystyle 0=n_{0}<n_{1}<n_{2}<\ldots <n_{q-1}<n_{q}=n,\qquad n_{i}=\dim \ker L^{i}.}
The signature characterizes
L
{\displaystyle L}
up to an invertible
linear transformation . Furthermore, it satisfies the inequalities
n
j
+
1
−
n
j
≤
n
j
−
n
j
−
1
,
for all
j
=
1
,
…
,
q
−
1.
{\displaystyle n_{j+1}-n_{j}\leq n_{j}-n_{j-1},\qquad {\mbox{for all }}j=1,\ldots ,q-1.}
Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.
Additional properties
Generalizations
A
linear operator
T
{\displaystyle T}
is locally nilpotent if for every vector
v
{\displaystyle v}
, there exists a
k
∈
N
{\displaystyle k\in \mathbb {N} }
such that
T
k
(
v
)
=
0.
{\displaystyle T^{k}(v)=0.\!\,}
For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.
Notes
^
Herstein (1975 , p. 294)
^
Beauregard & Fraleigh (1973 , p. 312)
^
Herstein (1975 , p. 268)
^
Nering (1970 , p. 274)
^ Mercer, Idris D. (31 October 2005).
"Finding "nonobvious" nilpotent matrices" (PDF) . idmercer.com . self-published; personal credentials: PhD Mathematics,
Simon Fraser University . Retrieved 5 April 2023 .
^
Beauregard & Fraleigh (1973 , p. 312)
^
Beauregard & Fraleigh (1973 , pp. 312, 313)
^ R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra , Vol. 56, No. 3
References
Beauregard, Raymond A.; Fraleigh, John B. (1973),
A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields , Boston:
Houghton Mifflin Co. ,
ISBN
0-395-14017-X
Herstein, I. N. (1975), Topics In Algebra (2nd ed.),
John Wiley & Sons
Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York:
Wiley ,
LCCN
76091646
External links