主理想环上一类矩阵对可同时三角化探讨

Study on Simultaneous Triangularization of a Class of Matrix Pair over the Ring of Principal Ideals

矩阵的三角化是矩阵论的重要组成部分。关于交换环上矩阵对可同时三角化的问题已有许多研究成果。为探索将矩阵可同时三角化问题引入到主理想环研究中,将主理想环上矩阵可同时三角化问题作为研究对象,借助二次最小多项式,得到了一类矩阵在主理想环上对可同时三角化的一个充分且必要条件,同时得到了通过有限步验证程序,将矩阵对化简为下三角矩阵的一种方法,推广了有关矩阵可三角化理论的研究。

The triangulation of matrices is an important part of matrix theory.There are many research results on the simultaneous triangulation of matrix pairs on?the commutative ring.In order to explore the application of matrix simultaneous triangulation to the main ideal ring,the problem of matrix simultaneous triangulation on the main ideal ring was taken as the research object;a sufficient and necessary condition for simultaneous triangulation of a class of matrices over principal ideal rings was obtained by means of quadratic minimal polynomials.Meanwhile,by finite step verification program,a method to simplify the matrix pair to the lower triangular matrix was obtained,which extends the research on the triangulation of matrices.