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The Expressions for the Generalized Inverses of Block Matrices

Author: XuHaiZuo
Tutor: SunLePing
School: Shanghai Normal University
Course: Computational Mathematics
Keywords: Generalized Inverse Partitioned Matrix Permutation matrix Full rank decomposition
CLC: O151.21
Type: Master's thesis
Year: 2008
Downloads: 231
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Abstract


As we all know , if M is a nonsingular matrix , then there exists a matrix G such that MG = GM = I, G is called the inverse matrix of the matrix M , denoted for M -1 . If the matrix M is a singular matrices or long matrix , inverse matrix G does not exist . E. H. Moore and R. Penrose extended the inverse matrix of symbols and the concept of a generalized inverse matrix . There are various forms of the generalized inverse of a matrix inside the Moore-Penrose generalized inverse addition to M ? , the rest are not unique . In addition , several important generalized inverse to including MP inverse M ? , the inverse weighted MP M X , Y ? group inverse M < sub > g and the Drain inverse M d are some sort with the specified range and null space M { 2 } the inverse M T S < sup> 2 , linear equations , differential equations , differential equations , optimal control has a wide range of applications . The expression of a variety of generalized inverse of this study , in order to simplify the problem , we have a matrix M block M to study the generalized inverse of its sub-block . For any matrix M ∈ C r m × n < / sup> by row permutation and column replacement can be one of the r -order non- singular matrix over the upper left corner , that exist permutation matrices P and Q, so that on the right side has a special form can be obtained by the expression of the generalized inverse of the arbitrary matrix M . Can only be replaced by the row or column permutation , to give M = P ( ? ) Or M = (AB) in the form of Q , where A is the rank of full rank matrix of R rows and columns of the array of full rank , which can be obtained Square block corresponding results. The first chapter describes some of the symbols and the definition of the generalized inverse used in later chapters . In Chapters II and III , through the replacement of the M is divided into four parts and two MP inverse , M T, S 2 and several special generalized inverse of the forms of expression , and several numerical example is given to verify the previous theorem . Can be seen by way of example , our results are valid .

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Theory of algebraic equations,linear algebra > Linear Algebra > Matrix theory
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