【作者】 黄绚晨；

【导师】 魏木生；

【作者基本信息】 华东师范大学 ， 计算数学， 2006， 硕士

【摘要】 本文主要研究了如下几类问题： 矩阵的偏序是当前矩阵论研究的一个热点，国内外许多学者从事矩阵偏序的研究，他们研究各种类型的矩阵偏序，并应用到数理统计等学科中。矩阵分解在矩阵理论中有着极其重要的作用。本文的主要工具是矩阵的奇异值(SVD)分解，本文从矩阵的偏序定义出发，提出了在集合意义下的新的矩阵广义逆偏序的定义，A≤{1}B(?)AA{1}=BA{1}，A{1}A=A{1}B以及A≤{1，2}B(?)AA{1，2}=BA{1，2}，A{1，2}A=A{1，2}B．并分别讨论了四种情况下，矩阵A，B的形式，最后得到了相应的广义逆偏序的充要条件。 同时本文还研究了在f(x)为一般多项式函数时解的情况，给出了矩阵函数方程可解的定义，利用矩阵函数的定义和性质以及矩阵的Jordan标准形理论，分别讨论了矩阵多项式函数方程f(X)=A在实数域和复数域上有解的充要条件，以及求解的方法步骤。此外还给出可以用A的多项式来表示方程的解的充要条件。更多还原

【Abstract】 This thesis mainly studies the following problems:Partial ordering of matrices is one of the most discussed points on the matrix theory. Many specialists have been engaged in studying the partial ordering of matrices such as varied kinds of partial ordering and its applications to mathematical statistics. SVD is one of the most important and widely tools in Matrix Analysis. By the definition of partial ordering of matrices, some new definitions of partial ordering have been put forward, such as A ≤{1} B(?) AA{1} = BA{1}, A{1}A = A{1}B and A ≤{1,2} B(?) AA{1, 2} = BA{1, 2}, A{1, 2}A = A{1, 2}B. It is discussed four situations in detail, and sufficient and necessary conditions of the new partial ordering have been derived.Meanwhile in this dissertation, we redefine the solvability of the equation of matrix polynomial function, when the function f(X)is a general polynomial function. Using the definition and property of matrix function as well as Jordan canonical form, we discuss necessary and sufficient conditions of matrix polynomial functionf(X) = A over the field of R and C respectively, and give a procedure to solve it. We also give necessary and sufficient conditions that any solution can be explicitly expressed by matrix polynomial of A.更多还原