有强法式的体上矩阵

Matrix Having Strong Normal Form over Division Ring

设D为除环,A∈Dn×n,则可用初等变换将λI-A化简为对角阵A= diag(1,…,1,φ1,…,φr),其中(?)i为D上首1多项式并且φ1|…|φr.如果这个对角阵A在形状上是唯一的,则称A是有强法式的矩阵.本文应用中心原子因子与初等因子给出了体上有强法式的矩阵的本质刻画,给出了体上矩阵有强法式的一些充要条件.

Let D be a division ring, A∈D^n×n. Then λI-A can be reduced by some elementary operations into a diagonal form Λ=diag（1,…1,φl,…φr）, where φi is a monic polynomial with φl｜…｜φr. A matrix A is called having strong normal form if the diagonal form Λ has a unique form. In this paper, the matrix having strong normal form is characterized by the center atom divisors and the elementary divisors, and some necessary and sufficient conditions for a matrix over D to have strong normal form are obtained.