for some positive integer. The smallest such is called the index of ,[1] 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 ).[2][3][4] Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.
Examplesedit
Example 1edit
The matrix
is nilpotent with index 2, since .
Example 2edit
More generally, any -dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index [citation needed]. For example, the matrix
is nilpotent, with
The index of is therefore 3.
Example 3edit
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 4edit
Additionally, any matrices of the form
such as
or
square to zero.
Example 5edit
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]
Example 6edit
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.
Characterizationedit
For an square matrix with real (or complex) entries, the following are equivalent:
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:
^Mercer, Idris D. (31 October 2005). "Finding "nonobvious" nilpotent matrices" (PDF). idmercer.com. self-published; personal credentials: PhD Mathematics, Simon Fraser University. Retrieved 5 April 2023.
^R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3
Referencesedit
Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X