In linear algebra, a nilpotent matrix is a square matrix N such that for some positive integer
[2][3][4] Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.
The matrix is nilpotent with index 2, since
-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index
For example, the matrix is nilpotent, with The index of
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.
Additionally, any matrices of the form such as or square to zero.
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.
[5] 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.
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.
Newton's identities) This theorem has several consequences, including: See also: Jordan–Chevalley decomposition#Nilpotency criterion.
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
is a shift matrix (possibly of different sizes).
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.
Furthermore, it satisfies the inequalities Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.
such that For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.