矩陣的分解

摘要

眾所周知，矩陣是代數(shù)學中的一個重要概念，它的出現(xiàn)促進了代數(shù)學的快速發(fā)展．矩陣分解作為矩陣理論中非常重要的一部分，是指將一個矩陣分解成一些特殊類型矩陣的乘積（或和）的形式．矩陣分解的內容豐富，形式多樣，是解決某些線性代數(shù)問題的重要工具．本文主要從矩陣的QR分解、滿秩分解、三角分解和奇異值分解等方面對矩陣的分解作了論述，首先給出了這幾種分解形式的定義以及相關性質，然后給出了它們各自的具體的分解方法，最后通過例題的形式將各分解方法呈現(xiàn)出來.

關鍵詞：矩陣；分解；QR分解；三角分解；滿秩分解

The Decomposition of the Matrix

ABSTRACT

As everyone knows，matrix is one of the most important concepts in algebra，whose appearance  promotes the development of algebra．While as a significant part of the theory of matrix，the decomposition of matrix aims at decomposing a matrix into the product(or sum) of several specific kinds of matrices．The decomposition of matrix not only concludes rich contents and forms，but also works as one of the significant methods in dealing with some linear algebra problems．In this paper，the decomposition of matrix is mainly introduced from the aspects mentioned below，such as QR decomposition，full rank decomposition，LU decomposition and so on．Firstly，the definitions and related properties of these forms of decomposition are given．And then，specific decomposition ways of theirs are illustrated．Finally，these decomposition methods are clearly presented by the forms of some examples．

Keywords：Matrix；Decomposition；QR Decomposition；LU Matrix Decomposition；Full Rank Decomposition

目錄

摘要 I

ABSTRACT II

目錄 III

一、引言 1

二、矩陣的QR分解 1

（一）矩陣QR分解的基本概念及定理 1

（二）矩陣QR分解的常用方法及應用舉例 1

三、矩陣的三角分解 8

（一）矩陣三角分解的基本概念及定理 8

（二）矩陣三角分解的常用方法及應用舉例 9

四、矩陣的滿秩分解 15

（一）矩陣滿秩分解的基本概念及定理 15

（二）矩陣滿秩分解的常用方法及應用舉例 15

五、矩陣的奇異值分解 17

（一）矩陣奇異值分解的基本概念及定理 17

（二）矩陣奇異值分解的常用方法及應用舉例 18

六、結論 20

參考文獻 20

致謝 21