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
(and thus,
for all
). 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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
(or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
) entries, the following are equivalent:
*
is nilpotent.
* The
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The char ...
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of characteristic 0 or sufficiently large characteristic. (cf.
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynom ...
)
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
(upper)
shift matrix:
:
This matrix has 1s along the
superdiagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Gre ...
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
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical exampl ...
nilpotent matrix.
Specifically, if
is any nilpotent matrix, then
is
similar to a
block diagonal matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original ma ...
of the form
:
where each of the blocks
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
:
That is, if
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. (It is not necessary for the field to be algebraically closed.)
Flag of subspaces
A nilpotent transformation
on
naturally determines a
flag of subspaces
:
and a signature
:
The signature characterizes
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
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.
Notes
References
*
*
*
External links
Nilpotent matrixan
nilpotent transformationon
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...
.
{{Matrix classes
Matrices