數學畢業論文-矩陣分解與矩陣方程AX=B，AXB=C的解

矩陣分解和矩陣方程AX=B，AXB=C的解

摘要

本文詳細地闡述了矩陣的QR分解，奇異值分解，利用這些矩陣分解，給出了矩陣方程AX=B， AXB=C有解條件及有解時解的.1般表示式。

關鍵詞：初等變換；QR分解；奇異值分解；矩陣方程。

Matrix Decomposition and the Solutions of the Matrix Equations AX=B,AXB=C

Abstract
In this paper,  the QR decomposition,  Singular Value decomposition of matrix have been remarked in detail.
Using these matrix decompositions,  the necessary and sufficient conditions of the matrix equations  AX=B  and
AXB=C  have a solution and have an unique solution are given.  And when the matrix equations have a Solution or  have an unique solution, the expression of the solution has been presented.

Key Words:  Fundamental transformation; QR decomposition; Singular value decomposition; matrix equation.

目   錄

1引言……………………………………………………………………………………………1
2矩陣的兩種分解………………………………………………………………………………1
  2.1  QR分解…………………………………………………………………………………2
2.2  奇異值分解………………………………………………………………………………2
3 AX=B的解法 …………………………………………………………………………………4
  3.1　矩陣 可逆……………………………………………………………………………6
  3.2  矩陣 為滿秩矩陣……………………………………………………………………7
  3.3   不可逆………………………………………………………………………………7
  3.4　典型題舉例……………………………………………………………………………7
4 AXB=C的解法 ………………………………………………………………………………12
 4.1  若 ， 均可逆…………………………………………………………………………14
  4.2  若 （ ）可逆， （ ）不可逆………………………………………………………14
  4.3  若 ， 均不可逆………………………………………………………………………14
  4.4  典型題舉例……………………………………………………………………………14
5結束語…………………………………………………………………………………………18
6參考文獻………………………………………………………………………………………19