矩陣的分解與應用

矩陣的分解與應用

摘要：矩陣分解在很多領域獲得了廣泛的應用。在數(shù)值代數(shù)中，利用矩陣分解可以將規(guī)模較大的復雜問題轉(zhuǎn)化為小規(guī)模的簡單的問題來求解；在應用統(tǒng)計領域，通過矩陣分解得到原數(shù)據(jù)矩陣的低秩逼近，從而發(fā)現(xiàn)數(shù)據(jù)的內(nèi)存接頭特征；在機器學習和模式識別的應用中，矩陣的.低秩逼近可以大大降低數(shù)據(jù)特征的維數(shù)，節(jié)省存儲和計算資源。本文著重研究了矩陣的ＬＵ分解和奇異值分解以及它們的應用，并用實例進行了計算，選擇了1些實際的例子來進1步了解矩陣分解在科技方面的應用。在1些可以用計算機程序處理問題的地方，加入了MATLAB軟件的計算過程，這使我們更快地解決了更為復雜的計算問題。
關(guān)鍵詞：低秩逼近；矩陣的LU分解；奇異值分解 ；應用

Matrix Decomposition and Its Application

Abstract: Matrix decomposition has been widely applied in many fields. In the numeral value algebra, we can adopt matrix decomposition to solve a problem by transforming the large-scale complicated problem into small-scale simple problems; in the applied statistics, we can get the low rank approximation of the original data matrix through matrix decomposition so as to discover the data’s internal connection. During the machinery’s study and the application of mode identification, the low rank approximation of matrix can largely lower dimensions of the data’s characteristic and conserve the storage and computation resources. This article mainly focused on LU Decomposition, Singular Value Decomposition and their application with examples. The practical examples helped us further understand the matrix decomposition’s application in science and technology field. The adding of MATLAB software computing to the computer procedures processing makes us solve the more complicated problems more quickly and more accurately.
Key words: the low rank approximation; LU Decomposition of matrix; SVD; application