# Nilpotent matrix

Jump to navigation Jump to search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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

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

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 Lk = 0 for some positive integer k (and thus, Lj = 0 for all jk). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

## Examples

The matrix

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

is nilpotent, since M2 = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent. For example, the matrix

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

is nilpotent, with

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

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

${\displaystyle N={\begin{bmatrix}5&-3&2\\15&-9&6\\10&-6&4\end{bmatrix}}}$

squares to zero, though the matrix has no zero entries.

## Characterization

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

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 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:

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

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 slot to the left:

${\displaystyle S(x_{1},x_{2},\ldots ,x_{n})=(x_{2},\ldots ,x_{n},0).}$

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

${\displaystyle {\begin{bmatrix}S_{1}&0&\ldots &0\\0&S_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &S_{r}\end{bmatrix}}}$

where each of the blocks S1S2, ..., Sr 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

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

That is, if 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 L on Rn 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}

The signature characterizes 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

${\displaystyle (I+N)^{-1}=I-N+N^{2}-N^{3}+\cdots ,}$
where only finitely many terms of this sum are nonzero.
• If N is nilpotent, then
${\displaystyle \det(I+N)=1,\!\,}$
where I denotes the n × n identity matrix. Conversely, if A is a matrix and
${\displaystyle \det(I+tA)=1\!\,}$
for all values of t, then A is nilpotent.

## Generalizations

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

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

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

## References

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