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The index of an [math]n \times n[/math] nilpotent matrix is always less than or equal to [math]n[/math]. As to your original problem, you know B^n = 0 for some n. Now suppose it were invertible and let C be it's inverse. The concept of a nilpotent matrix can be generalized to that of a nilpotent operator. By Nilpotent matrix, we mean any matrix A such that A^m = 0 where m can be any specific integer. Theorem (Characterization of nilpotent matrices). This means that there is an index k such that Bk = O. Nilpotent operator. In general, sum and product of two nilpotent matrices are not necessarily nilpotent. This definition can be applied in particular to square matrices.The matrix = is nilpotent because A 3 = 0. Examples. A^m=0 may be true for just m=3 but not for m=1 or m=2. For example, every [math]2 \times 2[/math] nilpotent matrix squares to zero. 0 0 0 3 0. Products of Nilpotent Matrices Pei Yuan Wu* Department of Applied Mathematics National Chiao Tung University Hsinchu, Taiwan, Republic of China Submitted by Thomas J. Laffey ABSTRACT We show that any complex singular square matrix T is a product of two nilpotent matrices A and B with rank A = rank B = rank T except when T is a 2 X 2 nilpotent matrix of rank one. 6 0 0 0 0. The determinant and trace of a nilpotent matrix are always zero. if p is the least positive integer for which A p = O, then A is said to be nilpotent of index p. (c) Periodic Matrix: But then 0 = CB^n = B^(n-1), a contradiction. nilpotent matrix The square matrix A is said to be nilpotent if A n = A ⁢ A ⁢ ⋯ ⁢ A ⏟ n times = 𝟎 for some positive integer n (here 𝟎 denotes the matrix where every entry is 0). I've tried various things like assigning the matrix to variable A then do a solve(A^X = 0) but I only get "warning solutions may have been lost" Hi, I have the following matrix and I have to find it's nilpotent index... 0 0 0 0 0. 0 2 0 0 0. I = I. Definition 2. 0 0 8 0 0. the index of the matrix (i.e., the smallest power after which null spaces stop growing). If, you still have problem in understanding then please feel free to write back. Consequently, a nilpotent matrix cannot be invertible. But if the two nilpotent matrices commute, then their sum and product are nilpotent as well. It does not mean that A^m=0 for every integer. In the factor ring Z/9Z, the equivalence class of 3 is nilpotent because 3 2 is congruent to 0 modulo 9.; Assume that two elements a, b in a ring R satisfy ab = 0.Then the element c = ba is nilpotent as c 2 = (ba) 2 = b(ab)a = 0. Then CB = I. of A.The off-diagonal entries of Tseem unpredictable and out of control. The matrix A would still be called Nilpotent Matrix. An n×n matrix B is called nilpotent if there exists a power of the matrix B which is equal to the zero matrix. Say B^n = 0 where n is the smallest positive integer for which this is true. We highly recommend revising the lecture on the minimal polynomial while having the previous proposition in mind. See nilpotent matrix for more.. Recall that the Core-Nilpotent Decomposition of a singular matrix Aof index kproduces a block diagonal matrix ∙ C 0 0 L ¸ similar to Ain which Cis non-singular, rank(C)=rank ¡ Ak ¢,and Lis nilpotent of index k.Isitpossible (b) Nilpotent Matrix: A nilpotent matrix is said to be nilpotent of index p, (p ∈ N), i f A p = O, A p − 1 ≠ O, \left( p\in N \right),\;\; if \;\;{{A}^{p}}=O,\,\,{{A}^{p-1}}\ne O, (p ∈ N), i f A p = O, A p − 1 = O, i.e. A nilpotent matrix cannot have an inverse.

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