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摘要
A family F of square matrices of the same order is called a quasi-commuting family if (AB - BA)C = C(AB - BA) for all A, B, C is an element of F where A, B, C need not be distinct. Let f(k)(x(1), x(2), ... , x(p)), (k = 1, 2, ... , r), be polynomials in the indeterminates x(1), x(2), ... , x(p) with coefficients in the complex field C, and let M(1), M(2), ... , M(r) be n x n matrices over C which are not necessarily distinct. 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 n x n matrices A(1), A(2), ... , A(p) over C, if {A(1), A(2), ... , A(p), M(1), M(2), ... , M(r)} is a quasi-commuting family, then F (A(1), A(2), ... , A(p)) = 0 implies that delta(F)(A(1), A(2), ... , A(p)) = 0.
- 出版日期2011-1-15
