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nilpotent matrix is always less than or equal to. n {\displaystyle n} For example, every. 2 × 2 {\displaystyle 2\times 2} 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.
Nilpotent. In mathematics, an element of a ring is called nilpotent if there exists some positive integer , called the index (or sometimes the degree), such that . The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras. [1]
In mathematics, a Lie algebra is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras. We write , and for all . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. The lower central series for Lie algebras is ...
The Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as N ij = δ i,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the ...
In mathematics, specifically linear algebra, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator in a unique way as the sum of two other linear operators which are simpler to understand. Specifically, one part is potentially diagonalisable and the other is nilpotent.
Nilpotent orbits are finite in number. The Zariski closure of a nilpotent orbit is a union of nilpotent orbits. Jacobson–Morozov theorem: over a field of characteristic zero, any nilpotent element e can be included into an sl 2-triple {e,h,f} and all such triples are conjugate by Z G (e), the centralizer of e in G.
The Volterra operator is the corresponding integral operator T on the Hilbert space L2 (0,1) given by. The operator T is not nilpotent: take f to be the function that is 1 everywhere and direct calculation shows that Tn f ≠ 0 (in the sense of L2) for all n. However, T is quasinilpotent. First notice that K is in L2 (X, m), therefore T is compact.
Nilpotent groups are called so because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group of nilpotence degree and an element , the function : defined by ():= [,] (where [,] = is the commutator of and ) is nilpotent in the sense that the th iteration of the function is trivial: () = for all in .