矩陣的分解與應(yīng)用
矩陣的分解與應(yīng)用
摘要：矩陣分解在很多領(lǐng)域獲得了廣泛的應(yīng)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(fù)雜問題轉(zhuǎn)化為小規(guī)模的簡(jiǎn)單的問題來(lái)求解；在應(yīng)用統(tǒng)計(jì)領(lǐng)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(jī)器學(xué)習(xí)和模式識(shí)別的應(yīng)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(chǔ)和計(jì)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(yīng)用，并用實(shí)例進(jìn)行了計(jì)算，選擇了1些實(shí)際的例子來(lái)進(jìn)1步了解矩陣分解在科技方面的應(yīng)用。在1些可以用計(jì)算機(jī)程序處理問題的地方，加入了MATLAB軟件的計(jì)算過程，這使我們更快地解決了更為復(fù)雜的計(jì)算問題。
關(guān)鍵詞：低秩逼近；矩陣的LU分解；奇異值分解 ；應(yīng)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
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