Nilpotent matrix

In linear algebra, a nilpotent matrix is a square matrix "N" such that
for some positive integer formula_2. The smallest such formula_2 is called the index of formula_4, sometimes the degree of formula_4.
More generally, a nilpotent transformation is a linear transformation formula_6 of a vector space such that formula_7 for some positive integer formula_2 (and thus, formula_9 for all formula_10). 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 formula_12.
Example 2.
More generally, any formula_13-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index formula_14. For example, the matrix
is nilpotent, with
The index of formula_17 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 formula_22 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.
Characterization.
For an formula_25 square matrix formula_4 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. (cf. Newton's identities)
This theorem has several consequences, including:
Classification.
Consider the formula_25 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:
This matrix is nilpotent with degree formula_13, and is the canonical nilpotent matrix.
Specifically, if formula_4 is any nilpotent matrix, then formula_4 is similar to a block diagonal matrix of the form
where each of the blocks formula_44 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 formula_4 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. (It is not necessary for the field to be algebraically closed.)
Flag of subspaces.
A nilpotent transformation formula_6 on formula_48 naturally determines a flag of subspaces
and a signature
The signature characterizes formula_6 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.
Generalizations.
A linear operator formula_53 is locally nilpotent if for every vector formula_54, there exists a formula_55 such that
For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.