A simple way about the corresponding solution of a class of repeated eigenvalue

To the constant coefficient homogeneous linear differential equationsdX[]dt=AX(1),if the eigenvalue of A is λ_i(i=1,…,r),its number of repetition is n_i≥1,the corresponding m_i elementary divisors are(λ-λ_i)~(k_(i1)),…,(λ-λ_i)~(k_(im_i)),k_(i1)+…+k_(im_i)=n_i,then it correspond equation(1) n_i linear independence solutions,its structure is like X(t)=(P_1(t),…,P_n(t))′e~(λ_it),the degree of polynomial is P_j(t)≤M_i-1,(M_i=max｛k_(i1),…,k_(im_i)｝).Because M_i is difficult to solve,this paper utilizes the characteristic of similar matrices and the Jordan's normal form,an upper bound of the degree of polynomial P~((i))_j(t) between M_i-1 and n_i-1 is found,making calculating more convenient and efficient.