Nilpotent

Nilpotent

Topics from Linear Algebra

Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017

14. Show that if a square matrix A is nilpotent, then all its eigenvalues are zeroes. In particular, a nilpotent matrix is always singular (detA=0) $ ( {\text{det }}A = 0) $ and its trace is zero. Recall that a matrix A is called nilpotent if Ap = 0 for some positive integer p.

Relation between leader–follower consensus control and feedback vertex sets

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Journal Information

Published in Advanced Robotics, 2023

Daiki Sugiyama, Shun-ichi Azuma, Ryo Ariizumi, Toru Asai

Statement (iii) can be obtained by showing the equivalence of the following six statements. Equation (23) holds, for all , for all , is a nilpotent matrix, is acyclic, contains an FVS of the network G as a subset of ,which are proven as follows. : Trivial from (21) and (22).: Trivial from Definition 3.1.: It is given by the following well-known property: for a square matrix A.: Since the diagonal elements of are all one, the matrix is nonnegative in the sense that all the elements are equal to or greater than 0. From the Perron-Frobenius theorem, it follows that (c) holds if and only if all the eigenvalues of are 0. On the other hand, the eigenvalues of any nilpotent matrix are all 0. These prove the equivalence between (c) and (d).: The following lemma [22] presents a well-known property for the existence of cycles.

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Topics from Linear Algebra

Relation between leader–follower consensus control and feedback vertex sets