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Concept
Nilpotent matrix
Summary
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.
The matrix
is nilpotent with index 2, since .
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.
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.
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.
For an square matrix with real (or complex) 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. (cf. Newton's identities)
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.
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