in linear algebra, a nilpotent matrix is a square matrix n such that :n^k = 0\, for some positive integer k. the smallest such k is sometimes called the degree of n. more generally, a nilpotent transformation is a linear transformation l of a vector space such that l'k&nbsp;=&nbsp;0 for some positive integer k (and thus, l'j&nbsp;=&nbsp;0 for all j â¥ k). both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings. ==examples== the matrix : m = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} is nilpotent, since m2&nbsp;=&nbsp;0. more generally, any triangular matrix with 0s along the main diagonal is nilpotent. for example, the matrix : n = \begin{bmatrix} 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{bmatrix} is nilpotent, with : n^2 = \begin{bmatrix} 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ n^3 = \begin{bmatrix} 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ n^4 = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}. though the examples above have a large number of zero entries, a typical nilpotent matrix does not. for example, the matrix : n = \begin{bmatrix} 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end{bmatrix} squares to zero, though the matrix has no zero entries. ==characterization== for an n&nbsp;&times;&nbsp;n square matrix n with real (or complex) entries, the following are equivalent: * n is nilpotent. * the minimal polynomial for n is Î»k for some positive integer k â¤ n. * the characteristic polynomial for n is Î»n. * the only (complex) eigenvalue for n is 0. * tr(nk) = 0 for all k > 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 degree of an n&nbsp;&times;&nbsp;n nilpotent matrix is always less than or equal to n. for example, every 2&nbsp;&times;&nbsp;2 nilpotent matrix squares to zero. * the determinant and trace of a nilpotent matrix are always zero. * the only nilpotent diagonalizable matrix is the zero matrix. ==classification== consider the n&nbsp;&times;&nbsp;n shift matrix: :s = \begin{bmatrix} 0 & 1 & 0 & \ldots & 0 \
Matrices