In linear algebra, a nilpotent matrix is a square matrix N such that
:<math>N^k = 0\,</math>
for some positive integer <math>k</math>. The smallest such <math>k</math> is called the index of <math>N</math>, sometimes the degree of <math>N</math>.
More generally, a nilpotent transformation is a linear transformation <math>L</math> of a vector space such that <math>L^k = 0</math> for some positive integer <math>k</math> (and thus, <math>L^j = 0</math> for all <math>j \geq k</math>). 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
:<math>
A = \begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}
</math>
is nilpotent with index 2, since <math>A^2 = 0</math>.
Example 2
More generally, any <math>n</math>-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index <math>\le n</math> . For example, the matrix
:<math>
B=\begin{bmatrix}
0 & 2 & 1 & 6\\
0 & 0 & 1 & 2\\
0 & 0 & 0 & 3\\
0 & 0 & 0 & 0
\end{bmatrix}
</math>
is nilpotent, with
:<math>
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 & 6\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
;\
B^4=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
</math>
The index of <math>B</math> is therefore 4.
Example 3
Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,
:<math>
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}
</math>
although the matrix has no zero entries.
Example 4
Additionally, any matrices of the form
:<math>
\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}</math>
such as
:<math>
\begin{bmatrix}
5 & 5 & 5 \\
6 & 6 & 6 \\
-11 & -11 & -11
\end{bmatrix}
</math>
or
:<math>\begin{bmatrix}
1 & 1 & 1 & 1 \\
2 & 2 & 2 & 2 \\
4 & 4 & 4 & 4 \\
-7 & -7 & -7 & -7
\end{bmatrix}
</math>
square to zero.
Example 5
Perhaps some of the most striking examples of nilpotent matrices are <math>n\times n</math> square matrices of the form:
:<math>\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}</math>
The first few of which are:
:<math>\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
</math>
These matrices are nilpotent but there are no zero entries in any powers of them less than the index.
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 <math>n \times n</math> square matrix <math>N</math> with real (or complex) entries, the following are equivalent:
- <math>N</math> is nilpotent.
- The characteristic polynomial for <math>N</math> is <math>\det \left(xI - N\right) = x^n</math>.
- The minimal polynomial for <math>N</math> is <math>x^k</math> for some positive integer <math>k \leq n</math>.
- The only complex eigenvalue for <math>N</math> 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 <math>n \times n</math> nilpotent matrix is always less than or equal to <math>n</math>. For example, every <math>2 \times 2</math> 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 <math>n \times n</math> (upper) shift matrix:
:<math>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}.</math>
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:
:<math>S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0).</math>
This matrix is nilpotent with degree <math>n</math>, and is the canonical nilpotent matrix.
Specifically, if <math>N</math> is any nilpotent matrix, then <math>N</math> is similar to a block diagonal matrix of the form
:<math> \begin{bmatrix}
S_1 & 0 & \ldots & 0 \\
0 & S_2 & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & S_r
\end{bmatrix} </math>
where each of the blocks <math>S_1,S_2,\ldots,S_r</math> is a shift matrix (possibly of different sizes). This form is a special case of the Jordan canonical form for matrices.
For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix
:<math> \begin{bmatrix}
0 & 1 \\
0 & 0
\end{bmatrix}. </math>
That is, if <math>N</math> is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b<sub>1</sub>, b<sub>2</sub> such that Nb<sub>1</sub> = 0 and Nb<sub>2</sub> = b<sub>1</sub>.
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 <math>L</math> on <math>\mathbb{R}^n</math> naturally determines a flag of subspaces
:<math> \{0\} \subset \ker L \subset \ker L^2 \subset \ldots \subset \ker L^{q-1} \subset \ker L^q = \mathbb{R}^n</math>
and a signature
:<math> 0 = n_0 < n_1 < n_2 < \ldots < n_{q-1} < n_q = n,\qquad n_i = \dim \ker L^i. </math>
The signature characterizes <math>L</math> up to an invertible linear transformation. Furthermore, it satisfies the inequalities
:<math> n_{j+1} - n_j \leq n_j - n_{j-1}, \qquad \mbox{for all } j = 1,\ldots,q-1. </math>
Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.
Additional properties
Generalizations
A linear operator <math>T</math> is locally nilpotent if for every vector <math>v</math>, there exists a <math>k\in\mathbb{N}</math> such that
:<math>T^k(v) = 0.\!\,</math>
For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.