# Nilpotent matrix

Short description: Mathematical concept in algebra

In linear algebra, a nilpotent matrix is a square matrix N such that

$\displaystyle{ N^k = 0\, }$

for some positive integer $\displaystyle{ k }$. The smallest such $\displaystyle{ k }$ is called the index of $\displaystyle{ N }$,[1] sometimes the degree of $\displaystyle{ N }$.

More generally, a nilpotent transformation is a linear transformation $\displaystyle{ L }$ of a vector space such that $\displaystyle{ L^k = 0 }$ for some positive integer $\displaystyle{ k }$ (and thus, $\displaystyle{ L^j = 0 }$ for all $\displaystyle{ j \geq k }$).[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

$\displaystyle{ A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} }$

is nilpotent with index 2, since $\displaystyle{ A^2 = 0 }$.

### Example 2

More generally, any $\displaystyle{ n }$-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index $\displaystyle{ \le n }$[citation needed]. For example, the matrix

$\displaystyle{ B=\begin{bmatrix} 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{bmatrix} }$

is nilpotent, with

$\displaystyle{ B^2=\begin{bmatrix} 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ B^3=\begin{bmatrix} 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ B^4=\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}. }$

The index of $\displaystyle{ B }$ is therefore 4.

### Example 3

Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,

$\displaystyle{ C=\begin{bmatrix} 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end{bmatrix} \qquad C^2=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} }$

although the matrix has no zero entries.

### Example 4

Additionally, any matrices of the form

$\displaystyle{ \begin{bmatrix} a_1 & a_1 & \cdots & a_1 \\ a_2 & a_2 & \cdots & a_2 \\ \vdots & \vdots & \ddots & \vdots \\ -a_1-a_2-\ldots-a_{n-1} & -a_1-a_2-\ldots-a_{n-1} & \ldots & -a_1-a_2-\ldots-a_{n-1} \end{bmatrix} }$

such as

$\displaystyle{ \begin{bmatrix} 5 & 5 & 5 \\ 6 & 6 & 6 \\ -11 & -11 & -11 \end{bmatrix} }$

or

$\displaystyle{ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \\ -7 & -7 & -7 & -7 \end{bmatrix} }$

square to zero.

### Example 5

Perhaps some of the most striking examples of nilpotent matrices are $\displaystyle{ n\times n }$ square matrices of the form:

$\displaystyle{ \begin{bmatrix} 2 & 2 & 2 & \cdots & 1-n \\ n+2 & 1 & 1 & \cdots & -n \\ 1 & n+2 & 1 & \cdots & -n \\ 1 & 1 & n+2 & \cdots & -n \\ \vdots & \vdots & \vdots & \ddots & \vdots \end{bmatrix} }$

The first few of which are:

$\displaystyle{ \begin{bmatrix} 2 & -1 \\ 4 & -2 \end{bmatrix} \qquad \begin{bmatrix} 2 & 2 & -2 \\ 5 & 1 & -3 \\ 1 & 5 & -3 \end{bmatrix} \qquad \begin{bmatrix} 2 & 2 & 2 & -3 \\ 6 & 1 & 1 & -4 \\ 1 & 6 & 1 & -4 \\ 1 & 1 & 6 & -4 \end{bmatrix} \qquad \begin{bmatrix} 2 & 2 & 2 & 2 & -4 \\ 7 & 1 & 1 & 1 & -5 \\ 1 & 7 & 1 & 1 & -5 \\ 1 & 1 & 7 & 1 & -5 \\ 1 & 1 & 1 & 7 & -5 \end{bmatrix} \qquad \ldots }$

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 $\displaystyle{ n \times n }$ square matrix $\displaystyle{ N }$ with real (or complex) entries, the following are equivalent:

• $\displaystyle{ N }$ is nilpotent.
• The characteristic polynomial for $\displaystyle{ N }$ is $\displaystyle{ \det \left(xI - N\right) = x^n }$.
• The minimal polynomial for $\displaystyle{ N }$ is $\displaystyle{ x^k }$ for some positive integer $\displaystyle{ k \leq n }$.
• The only complex eigenvalue for $\displaystyle{ N }$ 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 $\displaystyle{ n \times n }$ nilpotent matrix is always less than or equal to $\displaystyle{ n }$. For example, every $\displaystyle{ 2 \times 2 }$ 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.

## Classification

Consider the $\displaystyle{ n \times n }$ (upper) shift matrix:

$\displaystyle{ S = \begin{bmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{bmatrix}. }$

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:

$\displaystyle{ S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0). }$[6]

This matrix is nilpotent with degree $\displaystyle{ n }$, and is the canonical nilpotent matrix.

Specifically, if $\displaystyle{ N }$ is any nilpotent matrix, then $\displaystyle{ N }$ is similar to a block diagonal matrix of the form

$\displaystyle{ \begin{bmatrix} S_1 & 0 & \ldots & 0 \\ 0 & S_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & S_r \end{bmatrix} }$

where each of the blocks $\displaystyle{ S_1,S_2,\ldots,S_r }$ 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

$\displaystyle{ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. }$

That is, if $\displaystyle{ N }$ is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b1b2 such that Nb1 = 0 and Nb2 = b1.

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 $\displaystyle{ L }$ on $\displaystyle{ \mathbb{R}^n }$ naturally determines a flag of subspaces

$\displaystyle{ \{0\} \subset \ker L \subset \ker L^2 \subset \ldots \subset \ker L^{q-1} \subset \ker L^q = \mathbb{R}^n }$

and a signature

$\displaystyle{ 0 = n_0 \lt n_1 \lt n_2 \lt \ldots \lt n_{q-1} \lt n_q = n,\qquad n_i = \dim \ker L^i. }$

The signature characterizes $\displaystyle{ L }$ up to an invertible linear transformation. Furthermore, it satisfies the inequalities

$\displaystyle{ n_{j+1} - n_j \leq n_j - n_{j-1}, \qquad \mbox{for all } j = 1,\ldots,q-1. }$

Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.

## Additional properties

• If $\displaystyle{ N }$ is nilpotent of index $\displaystyle{ k }$ , then $\displaystyle{ I+N }$ and $\displaystyle{ I-N }$ are invertible, where $\displaystyle{ I }$ is the $\displaystyle{ n \times n }$ identity matrix. The inverses are given by
\displaystyle{ \begin{align} (I + N)^{-1} &= \displaystyle\sum^k_{m=0}\left(-N\right)^m = I - N + N^2 - N^3 + N^4 - N^5 + N^6 - N^7 + \cdots +(-N)^k \\ (I - N)^{-1} &= \displaystyle\sum^k_{m=0}N^m = I + N + N^2 + N^3 + N^4 + N^5 + N^6 + N^7 + \cdots + N^k \\ \end{align} }
• If $\displaystyle{ N }$ is nilpotent, then
$\displaystyle{ \det (I + N) = 1. }$

Conversely, if $\displaystyle{ A }$ is a matrix and

$\displaystyle{ \det (I + tA) = 1\!\, }$
for all values of $\displaystyle{ t }$, then $\displaystyle{ A }$ is nilpotent. In fact, since $\displaystyle{ p(t) = \det (I + tA) - 1 }$ is a polynomial of degree $\displaystyle{ n }$, it suffices to have this hold for $\displaystyle{ n+1 }$ distinct values of $\displaystyle{ t }$.
• Every singular matrix can be written as a product of nilpotent matrices.[8]
• A nilpotent matrix is a special case of a convergent matrix.

## Generalizations

A linear operator $\displaystyle{ T }$ is locally nilpotent if for every vector $\displaystyle{ v }$, there exists a $\displaystyle{ k\in\mathbb{N} }$ such that

$\displaystyle{ T^k(v) = 0.\!\, }$

For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.

## Notes

1. (Herstein 1975)
2. (Beauregard Fraleigh)
3. (Herstein 1975)
4. (Nering 1970)
5. Mercer, Idris D. (31 October 2005). "Finding "nonobvious" nilpotent matrices". self-published; personal credentials: PhD Mathematics, Simon Fraser University.
6. (Beauregard Fraleigh)
7. (Beauregard Fraleigh)
8. R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3