互素多项式在矩阵秩中的应用

Applications of Coprime Polynomials to Ranks of Matrixes

给出了互素多项式在矩阵秩讨论中的几个结果:1)设f(x),g(x)∈P[x],A∈Mn(P).若f(x),g(x)互素,且f(A)g(A)=0,则r(f(A))+r(g(A))=n.2)设fi(x)∈P[x],i=1,2,…,m,A∈Mn(P).若f1(x),f2(x),…,fm(x)互素,且f1(A)f2(A)…fm(A)=0,则n≤r(f1(A))+r(f2(A))+…+r(fm(A))≤(m-1)n.3)设fi(x)∈P[x],i=1,2,…,m,A∈Mn(P),若f1(x),f2(x),…,fm(x)两两互素,且fi(A)fj(A)=0,i≠j,i,j=1,2,…,m,则r(f1(A))+r(f2(A))+…+r(fm(A))=n.

In this paper,some results on applications of coprime polynomials to ranks of matrixes are given as follows:1) Let f(x),g(x)∈P[x] and A∈M_n(P). If f(x),g(x) are coprime and f(A)g(A)=0, then (r (f(A))+r(g(A))=n).2) Let f_i(x)∈P[x],i=1,2,…,m and A∈M_n(P). If f_1(x),f_2(x),…,f_m(x) are coprime and f_1(A)f_2(A)…f_m(A)=0, then n≤r(f_1(A))+r(f_2(A))+…+r(f_m(A))≤(m-1)n.3) Let f_i(x)∈P[x],i=1,2,…,m and A∈M_n(P). If f_i(x),f_j(x) are coprime and f_i(A)f_j(A)=0, for distinct i and j and i,j=1,2,…,m, thenr(f_1(A))+r(f_2(A))+…+r(f_m(A))=n.