Nilpotent Matrix

In linear algebra, a nilpotent matrix is a square matrix ''N'' such that
:$N^k = 0\,$
for some positive integer $k$. The smallest such $k$ is called the index of $N$, sometimes the degree of $N$.

More generally, a nilpotent transformation is a linear transformation $L$ of a vector space such that $L^k = 0$ for some positive integer $k$ (and thus, $L^j = 0$ for all $j \geq k$). 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
:$A = \begin 0 & 1 \\ 0 & 0 \end$
is nilpotent with index 2, since $A^2 = 0$.

Example 2

More generally, any $n$-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index $\le n$ . For example, the matrix
:$B=\begin 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end$
is nilpotent, with
:$B^2=\begin 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end ;\ B^3=\begin 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end ;\ B^4=\begin 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end.$

The index of $B$ is therefore 4.

Example 3

Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,
:$C=\begin 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end \qquad C^2=\begin 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end$
although the matrix has no zero entries.

Example 4

Additionally, any matrices of the form

:$\begin a_1 & a_1 & \cdots & a_1 \\ a_2 & a_2 & \cdots & a_2 \\ \vdots & \vdots & \ddots & \vdots \\ -a_1-a_2-\ldots-a_ & -a_1-a_2-\ldots-a_ & \ldots & -a_1-a_2-\ldots-a_ \end$

such as

:$\begin 5 & 5 & 5 \\ 6 & 6 & 6 \\ -11 & -11 & -11 \end$

or

:$\begin 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \\ -7 & -7 & -7 & -7 \end$

square to zero.

Example 5

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

:$\begin 2 & 2 & 2 & \cdots & 1-n \\ n+2 & 1 & 1 & \cdots & -n \\ 1 & n+2 & 1 & \cdots & -n \\ 1 & 1 & n+2 & \cdots & -n \\ \vdots & \vdots & \vdots & \ddots & \vdots \end$

The first few of which are:

:$\begin 2 & -1 \\ 4 & -2 \end \qquad \begin 2 & 2 & -2 \\ 5 & 1 & -3 \\ 1 & 5 & -3 \end \qquad \begin 2 & 2 & 2 & -3 \\ 6 & 1 & 1 & -4 \\ 1 & 6 & 1 & -4 \\ 1 & 1 & 6 & -4 \end \qquad \begin 2 & 2 & 2 & 2 & -4 \\ 7 & 1 & 1 & 1 & -5 \\ 1 & 7 & 1 & 1 & -5 \\ 1 & 1 & 7 & 1 & -5 \\ 1 & 1 & 1 & 7 & -5 \end \qquad \ldots$

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 $n \times n$ square matrix $N$ with real (or complex) entries, the following are equivalent:
* $N$ is nilpotent.
* The characteristic polynomial for $N$ is $\det \left(xI - N\right) = x^n$.
* The minimal polynomial for $N$ is $x^k$ for some positive integer $k \leq n$.
* The only complex eigenvalue for $N$ 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 $n \times n$ nilpotent matrix is always less than or equal to $n$. For example, every $2 \times 2$ 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 $n \times n$ (upper) shift matrix:
:$S = \begin 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end.$
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:
:$S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0).$
This matrix is nilpotent with degree $n$, and is the canonical nilpotent matrix.

Specifically, if $N$ is any nilpotent matrix, then $N$ is similar to a block diagonal matrix of the form
:$\begin S_1 & 0 & \ldots & 0 \\ 0 & S_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & S_r \end$
where each of the blocks $S_1,S_2,\ldots,S_r$ is a shift matrix (possibly of different sizes). This form is a special case of the Jordan canonical form for matrices.

For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix
:$\begin 0 & 1 \\ 0 & 0 \end.$
That is, if $N$ is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b₁, b₂ such that ''N''b₁ = 0 and ''N''b₂ = 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 $L$ on $\mathbb^n$ naturally determines a flag of subspaces
:$\ \subset \ker L \subset \ker L^2 \subset \ldots \subset \ker L^ \subset \ker L^q = \mathbb^n$
and a signature
:$0 = n_0 < n_1 < n_2 < \ldots < n_ < n_q = n,\qquad n_i = \dim \ker L^i.$

The signature characterizes $L$ up to an invertible linear transformation. Furthermore, it satisfies the inequalities
:$n_ - n_j \leq n_j - n_, \qquad \mbox j = 1,\ldots,q-1.$
Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.

Additional properties

Generalizations

A linear operator $T$ is locally nilpotent if for every vector $v$, there exists a $k\in\mathbb$ such that
:$T^k(v) = 0.\!\,$
For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.

Notes

References

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External links

Nilpotent matrix
an
nilpotent transformation
on PlanetMath.

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