分块矩阵的简单应用.doc

分块矩阵的简单应用 摘 要 矩阵是高等代数中的一个重要概念，也是高等数学很多分支研究问题的工具。而把一个比较大的矩阵分成若干子块，构成分块矩阵是处理矩阵问题的重要技巧。分块矩阵思想来源于对矩阵运算复杂度及存储空间的考虑。特别当矩阵太大不适合存储在计算机内存中的时候，通过分块矩阵允许计算机每次只处理存储在内存中几个子矩阵，支持向量传输结构的向量计算机能够更加高效地运行支持分块矩阵的矩阵算法。分块矩阵可以降低矩阵的阶数，使矩阵更加条理清晰，使得矩阵的相关运算简单化，并使矩阵证明方面的相关问题得以便捷的解决。本文重点就分块矩阵的定义、分块方法、基本运算，行列式和求逆矩阵的计算，以及关于矩阵的秩的方面的证明问题进行了分析。使用了大量的例题说明了分块矩阵的技巧可以使高等代数中的很多计算与证明问题简单化。所以了解分析并掌握分块矩阵的性质与应用及相关的技巧是非常必要的。 关键词：矩阵；分块矩阵；子矩阵 ABSTRACT Matrix is an important concept in high algebra, but also an instrument for research of many filiation in high algebra. And the means of dividing a large-scale matrix up into some small one is a main skill to resolve the question of matrix. The idea of partitioned matrix comes of the advisement to the complexity of matrixs calculate and the unit of space.Especially,when the matrix is too large to save in the EMS memory, the computer which support the network management vector transport can take order with the partitioned matrix algorithm in high efficiency, with the partitioned matrix permit the computer only deal with the submatrix that store in the EMS memory every time.Theory about block matrix could be used to decline high order matrix and make it clearer and easier to calculate and prove some problems about matrix. This paper focuses on the problems of the concept of block matrix, and the numeration of square matrixand the proof of matrix. It shows the convenience of the block matrix in the problems of matrix and high algebra by making use of a number of examples. It is necessary that we must learn and analyse and grasp the skill of block matrix which is an important concept in high algebra. Keywords： matrix；block matrix ；submatrix 目录 目录 III 1. 绪论 1 2. 分块矩阵的基本概念 4 2.1 定义 4 2.2 分块矩阵的运算 4 2.2.1加法与数量乘法运算 4 2.2.2乘法运算 5 2.2.3转置运算 5 2.3分块矩阵的初等变换 6 2.3.1定义 6 2.3.2运算 6 3. 分块矩阵在计算方面的应用 10 3.1 行列式计算 10 3.1.1 引理 10 3.1.2 定理及几条推论 10 3.2 求逆矩阵 15 3.2.1 块对角矩阵 15 3.2.2 一般的非奇异矩阵的逆矩阵 17 3.3解线性方程组 20 3.3.1齐次线性方程组 20 3.3.2非齐次线性方程组 22 4. 分块矩阵在证明方面的应用 24 4.1有关矩阵的秩的证明 24 结论 27