0 0 0 3 0. If, you still have problem in understanding then please feel free to write back. By Nilpotent matrix, we mean any matrix A such that A^m = 0 where m can be any specific integer. 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. The matrix A would still be called Nilpotent Matrix. the index of the matrix (i.e., the smallest power after which null spaces stop growing). Theorem (Characterization of nilpotent matrices). This definition can be applied in particular to square matrices.The matrix = is nilpotent because A 3 = 0. But if the two nilpotent matrices commute, then their sum and product are nilpotent as well. The determinant and trace of a nilpotent matrix are always zero. As to your original problem, you know B^n = 0 for some n. For example, every [math]2 \times 2[/math] nilpotent matrix squares to zero. Now suppose it were invertible and let C be it's inverse. of A.The off-diagonal entries of Tseem unpredictable and out of control. The index of an [math]n \times n[/math] nilpotent matrix is always less than or equal to [math]n[/math]. 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 Consequently, a nilpotent matrix cannot be invertible. 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). 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. See nilpotent matrix for more.. A^m=0 may be true for just m=3 but not for m=1 or m=2. (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. The concept of a nilpotent matrix can be generalized to that of a nilpotent operator. 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. We highly recommend revising the lecture on the minimal polynomial while having the previous proposition in mind. 6 0 0 0 0. 0 2 0 0 0. if p is the least positive integer for which A p = O, then A is said to be nilpotent of index p. 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