Nilpotent matrix

In linear algebra, a nilpotent matrix is a square matrix N such that
for some positive integer. The smallest such is called the index of, sometimes the degree of.
More generally, a nilpotent transformation is a linear transformation of a vector space such that for some positive integer . 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
is nilpotent with index 2, since.

Example 2

More generally, any -dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index . For example, the matrix
is nilpotent, with
The index of is therefore 4.

Example 3

Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example,
although the matrix has no zero entries.

Example 4

Additionally, any matrices of the form
such as
or
square to zero.

Example 5

Perhaps some of the most striking examples of nilpotent matrices are square matrices of the form:
The first few of which are:
These matrices are nilpotent but there are no zero entries in any powers of them less than the index.

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

For an square matrix with real entries, the following are equivalent:

is nilpotent.
The characteristic polynomial for is.
The minimal polynomial for is for some positive integer.
The only complex eigenvalue for is 0.

The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic.
This theorem has several consequences, including:

The index of an nilpotent matrix is always less than or equal to. For example, every 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 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 position to the left, with a zero appearing in the last position:
This matrix is nilpotent with degree, and is the canonical nilpotent matrix.
Specifically, if is any nilpotent matrix, then is similar to a block diagonal matrix of the form
where each of the blocks is a shift matrix. 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
That is, if 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.

Flag of subspaces

A nilpotent transformation on naturally determines a flag of subspaces
and a signature
The signature characterizes 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

Generalizations

A linear operator is locally nilpotent if for every vector, there exists a such that
For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.