Nilpotent matrix

Template:Short description 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>,<ref>Template:Harvtxt</ref> 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>).<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref> 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> Template:Citation needed. 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.<ref name="Mercer2005">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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

{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Ambox }} }} 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><ref>Template:Harvtxt</ref>

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.<ref>Template:Harvtxt</ref>

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 b₁, b₂ 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 <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.

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.

Notes

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References

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External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.