# Nilpotent matrix

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 index of N.[1]

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).[2][3][4] 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, with degree ${\displaystyle \leq n}$. 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

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

Although 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. Consequently, a nilpotent matrix cannot be invertible.
• 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 position to the left, with a zero appearing in the last position:

${\displaystyle S(x_{1},x_{2},\ldots ,x_{n})=(x_{2},\ldots ,x_{n},0).}$[5]

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 form is a special case of the Jordan canonical form for matrices.[6]

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.

${\displaystyle (I+N)^{-1}=I-N+N^{2}-N^{3}+\cdots ,}$
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. In fact, since ${\displaystyle p(t)=\det(I+tA)-1}$ is a polynomial of degree ${\displaystyle n}$, it suffices to have this hold for ${\displaystyle n+1}$ distinct values of ${\displaystyle t}$.

## 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.

## Notes

1. ^ Herstein (1975, p. 294)
2. ^ Beauregard & Fraleigh (1973, p. 312)
3. ^ Herstein (1975, p. 268)
4. ^ Nering (1970, p. 274)
5. ^ Beauregard & Fraleigh (1973, p. 312)
6. ^ Beauregard & Fraleigh (1973, pp. 312,313)
7. ^ R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3