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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 , [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 ).
Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with n = 2 {\displaystyle n=2} ). Both are linked, also through supersymmetry and Morse theory , [ 6 ] as shown by Edward Witten in a celebrated article.
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.
Nilpotent matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group.From the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes …, where is a partition of n.
A nilpotent element is an element a such that a n = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor. An idempotent is an element such that e 2 = e. One example of an idempotent element is a projection in linear algebra.
Write the triangular matrix U as U = D + N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains the eigenvalues of A in arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of A , while the Frobenius norm of A , squared, is the ...
A somewhat easier equivalent condition for the nilpotency of : is nilpotent if and only if is nilpotent (as a Lie algebra). To see this, first observe that (1) implies that a d g {\displaystyle \mathrm {ad} \,{\mathfrak {g}}} is nilpotent, since the expansion of an ( n − 1) -fold nested bracket will consist of terms of the form in (1).
The operator T is not nilpotent: take f to be the function that is 1 everywhere and direct calculation shows that T n f ≠ 0 (in the sense of L 2) for all n. However, T is quasinilpotent. First notice that K is in L 2 (X, m), therefore T is compact. By the spectral properties of compact operators, any nonzero λ in σ(T) is an eigenvalue.