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# 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 sometimes called 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* ≥ *k*). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Examples

The matrix

\( M = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \)

is nilpotent, since *M*^{2} = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent. For example, the matrix

\( N = \begin{bmatrix} 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{bmatrix} \)

is nilpotent, with

\( N^2 = \begin{bmatrix} 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ N^3 = \begin{bmatrix} 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ N^4 = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}. \)

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

\( \begin{bmatrix} 6 & -9 \\ 4 & -6 \end{bmatrix}\qquad\text{and}\qquad \begin{bmatrix} 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end{bmatrix} \)

both square to zero, though neither matrix has zero entries.

Characterization

For an *n* × *n* square matrix *N* with real (or complex) entries, the following are equivalent:

*N*is nilpotent.- The minimal polynomial for
*N*is λ^{k}for some positive integer*k*≤*n*. - The characteristic polynomial for
*N*is λ^{n}. - The only (complex) eigenvalue for
*N*is 0. - tr(N
^{k}) = 0 for all*k*> 0.

The last theorem holds true for matrices over any field of characteristic 0.

This theorem has several consequences, including:

- The degree of an
*n*×*n*nilpotent matrix is always less than or equal to*n*. For example, every 2 × 2 nilpotent matrix squares to zero. - The determinant and trace of a nilpotent matrix are always zero.
- The only nilpotent diagonalizable matrix is the zero matrix.

Classification

Consider the n × n shift matrix:

\( S = \begin{bmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{bmatrix}. \)

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 slot to the left:

\( S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0). \)

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

\( \begin{bmatrix} S_1 & 0 & \ldots & 0 \\ 0 & S_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & S_r \end{bmatrix} \)

where each of the blocks *S*_{1}, *S*_{2}, ..., *S*_{r} is a shift matrix (possibly of different sizes). This theorem is a special case of the Jordan canonical form for matrices.

For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix

\( \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. \)

That is, if *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 on Rn naturally determines a flag of subspaces

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

and a signature

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

The signature characterizes L up to an invertible linear transformation. Furthermore, it satisfies the inequalities

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

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

Additional properties

If N is nilpotent, then I + N is invertible, where I is the n × n identity matrix. The inverse is given by

\( (I + N)^{-1} = I - N + N^2 - N^3 + \cdots, \)

where only finitely many terms of this sum are nonzero.

If N is nilpotent, then

\( \det (I + N) = 1,\!\, \)

where I denotes the n × n identity matrix. Conversely, if A is a matrix and

\( \det (I + tA) = 1\!\, \)

for all values of t, then A is nilpotent.

Every singular matrix can be written as a product of nilpotent matrices.[1]

Generalizations

A linear operator T is locally nilpotent if for every vector v, there exists a k such that

\( T^k(v) = 0.\!\, \)

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

References

^ R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3

External links

Nilpotent matrix and nilpotent transformation on PlanetMath.

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