Nilpotent matrix

 ( Matrices ) 

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

$N^k\; =\; 0\backslash ,$

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 $j\; \backslash geq\; k$). Both of these concepts are special cases of a more general concept of nilpotent that applies to elements of rings.

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Examples

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Example 1 The matrix

$$

A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} is nilpotent with index 2, since $A^2\; =\; 0$.

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Example 2 More generally, any $n$-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index $\backslash le\; n$ . For example, the matrix

$$

B=\begin{bmatrix} 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{bmatrix} is nilpotent, with

$$

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 $B$ is therefore 4.

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Example 3 Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,

$$

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.

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Example 4 Additionally, any matrices of the form

$$

\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

$$

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

or

$\backslash begin\{bmatrix\}$

1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \\ -7 & -7 & -7 & -7 \end{bmatrix}

square to zero.

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Example 5 Perhaps some of the most striking examples of nilpotent matrices are $n\backslash times\; n$ square matrices of the form:

$\backslash begin\{bmatrix\}$

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:

$\backslash begin\{bmatrix\}$

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.

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Example 6 Consider the linear space of 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.

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Characterization For an $n\; \backslash times\; n$ square matrix $N$ with real number (or complex number) entries, the following are equivalent:

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

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Classification Consider the $n\; \backslash times\; n$ (upper) shift matrix:

$S\; =\; \backslash begin\{bmatrix\}$

  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:

$S(x\_1,x\_2,\backslash ldots,x\_n)\; =\; (x\_2,\backslash ldots,x\_n,0).$

This matrix is nilpotent with degree $n$, and is the Canonical form nilpotent matrix.

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

$\backslash begin\{bmatrix\}$

  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 $S\_1,S\_2,\backslash ldots,S\_r$ 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

$\backslash begin\{bmatrix\}$

  0 & 1 \\
  0 & 0
     

\end{bmatrix}. That is, if $N$ is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b₁,  b₂ such that N b ₁ = 0 and Nb₂ =  b₁.

This classification theorem holds for matrices over any field. (It is not necessary for the field to be algebraically closed.)

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Flag of subspaces A nilpotent transformation $L$ on $\backslash mathbb\{R\}^n$ naturally determines a flag of subspaces

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

and a signature

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

$n\_\{j+1\}\; -\; n\_j\; \backslash leq\; n\_j\; -\; n\_\{j-1\},\; \backslash qquad\; \backslash mbox\{for\; all\; \}\; j\; =\; 1,\backslash ldots,q-1.$

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

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Additional properties

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

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

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

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Notes

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External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.

Categories: Matrices

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