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. Examples Example 1 The matrix is nilpotent with index 2, since . Example 2 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. 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 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 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. Consequently, a nilpotent matrix cannot be invertible. The only nilpotent diagonalizable matrix is the zero matrix. See also: Jordan–Chevalley decomposition#Nilpotency criterion. Classification Consider the (upper) shift matrix: This matrix has 1s along the superdiagonal and 0s everywhere else. As a linear