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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.
The impulse analysis of the T-S fuzzy singular system via Kronecker index
Published in International Journal of Systems Science, 2019
Zhenghong Jin, Qingling Zhang, Xinyou Meng
For any matrix and and which the diagonal elements of their corresponding positions are zero at the same time. There exist a row transformation matrix P and a column transformation matrix Q such that where the matrix N is a nilpotent matrix and all diagonal elements are 0, is a n-dimensional lower triangular matrix in which all diagonal elements are 1.
Relation between leader–follower consensus control and feedback vertex sets
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.