# Nilpotent matrix

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

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

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

is nilpotent, with

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

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:

*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 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 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:

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:

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

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

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 **R**^{n} naturally determines a flag of subspaces

and a signature

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

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

- If
*N*is nilpotent, then

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

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