Wikipedia / Nilpotent matrix
Wikipedia / Nilpotent matrix
In linear algebra, a nilpotent matrix is a square matrix N such that
for some positive integer is called the index, sometimes the degree.
More generally, a nilpotent transformation is a linear transformation of a vector space). 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
.
Example 2
-dimensional triangular matrix with zeros along the main diagonal. For example, the matrix
is nilpotent, with
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
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
with real (or complex) entries, the following are equivalent:
is nilpotent.
The minimal polynomial.
is 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:
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
(upper) 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:
, and is the canonical nilpotent matrix.
is similar to a block diagonal matrix of the form
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 22 nilpotent matrix is similar to the matrix
is any nonzero 22 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. (It is not necessary for the field to be algebraically closed.)
Flag of subspaces
naturally determines a flag of subspaces
and a signature
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 such that
For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.