A generalized Cayley-Hamilton theorem

作者：Feng, Lianggui; Tan, Haijun; Zhao, Kaiming^*

来源：Linear Algebra and Its Applications, 2012, 436(7): 2440-2445.

DOI：10.1016/j.laa.2011.12.015

• Nilpotent matrix
• Cayley-Hamilton theorem

Let a be a solvable Lie subalgebra of the Lie algebra gl(n) (C) (= C-nxn as a vector space). Let f(k) (x(1), x(2), ..., x(p)), (k = 1, 2, ..., r), be polynomials in the commuting variables x(1), x(2), ..., x(p) with coefficients in C. For n x n matrices M-1, M-2, ..., M-r, let F(x(1), x(2), ..., x(p)) = Sigma(r)(k=1) M(k)f(k) (x(1), x(2), ..., x(p)) and let
delta F(x(1), x(2), ..., x(p)) = det F(x(1). x(2), ..., x(p)).
In this paper, we prove that, for A(1), A(2), ..., Ap, M-1, M-2, ..., M-r is an element of 3, if one value of the matrix-valued function F(A(1), A(2), ..., A(p)) (the value depends on the product order of the variables) is nilpotent, then, (a) all values of F(A(1), A(2), ..., Ap) are nilpotent; (b) all values of delta F(A(1), A(2), ..., A(p)) (again depends on the product order of the variables) are nilpotent, and one value is 0. This generalizes the recent result in [7] and makes his result accurate. The main tool we use in this paper is the representation theory of solvable Lie algebras.

• 单位
  中国人民解放军国防科学技术大学