矩阵的分解与应用

矩阵的分解与应用

摘要：矩阵分解在很多领域获得了广泛的应用。在数值代数中，利用矩阵分解可以将规模较大的复杂问题转化为小规模的简单的问题来求解；在应用统计领域，通过矩阵分解得到原数据矩阵的低秩逼近，从而发现数据的内存接头特征；在机器学习和模式识别的应用中，矩阵的低秩逼近可以大大降低数据特征的维数，节省存储和计算资源。本文着重研究了矩阵的ＬＵ分解和奇异值分解以及它们的应用，并用实例进行了计算，选择了1些实际的例子来进1步了解矩阵分解在科技方面的应用。在1些可以用计算机程序处理问题的地方，加入了MATLAB软件的计算过程，这使我们更快地解决了更为复杂的计算问题。

关键词：低秩逼近；矩阵的LU分解；奇异值分解 ；应用

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