Group Inverse for a Class of. Centrosymmetric Matrix

Transcription

1 International athematical Forum, Vol. 13, 018, no. 8, HIKARI Ltd, Group Inverse for a Class of Centrosymmetric atrix ei Wang and Junqing Wang Department of athematics, ianjin Polytechnic University ianjin, China, Copyright 018 ei Wang and Junqing Wang. his article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper a class of centrosymmetric matrix is considered, its full rank decomposition is based on, and the group inverse for this decomposition is obtained, and the relationship between mother matrix and the group inverse for this class of centrosymmetric matrix is derived. he results show that the method can not only reduce the calculated amount and memory space, but also can not reduce the numerical accuracy; so extend the results of the related references and spread its application scope. Keywords: centrosymmetric matrix; full rank decomposition; group inverse 1. Introduction here are a lot of centrosymmetric matrix in linear system theory, numerical analysis and other fields, therefore, the study of symmetric matrix is of great importance. Full rank decomposition is the most commonly used method in matrix decomposition, and it plays important roles in finding the generalized inverse of the matrix. Scholars have studied the properties, invertibility and inverse problems of centrosymmetric matrices, but, based on the full rank decomposition of a class of special centrosymmetric matrices, this paper studies the group inverse of matrices on the basis of full rank decomposition and obtains

2 35 ei Wang and Junqing Wang the relationship between the group inverse and the mother matrix., and generalizes and verifies the conclusions in the literature. nn Definition 1 If m ij satisfies mij mn 1 i, n1 j, i, j 1,,, n, then is called the centrosymmetric matrix. Lemma 1 [3] According to the definition 1, we have (1) he m-order centrosymmetric matrix can be expressed as A JB BJ, JAJ () he m+1- order centrosymmetric matrix can be expressed as A u BJ, JB Ju JAJ v v J Where ABis, an m -order square matrix, uvare, m -dimensional column vector, is an arbitrary number, J is a m -order square matrix with sub-diagonal elements of 1 and the remaining elements of 0. In particular, according to the lemma 1, if A B, uvare, m -dimensional zero vector, 0, then (1) he m-order centrosymmetric matrix can be expressed as A JA AJ, JAJ () he m+1- order centrosymmetric matrix can be expressed as A JA 0 AJ, JAJ he matrix can be regarded as the extended matrix of matrix A, and A is its mother matrix. Lemma [] mm Let the full rank decomposition of A is A FG, where F mr is column full rank decomposition, G rn

3 Group inverse for a class of centrosymmetric matrix 353 ranka rankf rankg r r 0, then A AJ F G GJ JA JAJ JF is the full rank decomposition of the m-order centrosymmetric matrix. Lemma 3 [] mm Let the full rank decomposition of A is A FG, where F mr is column full rank decomposition, G rn ranka rankf rankg r r 0, then A AJ F 0 G GJ JA JAJ JF is the full rank decomposition of the m+1- order centrosymmetric matrix.. heorem Lemma 4 [1] mm Let the full rank decomposition of A is A FG, A exists if and only if FG is reversible, then Proof: Let 1 A = F GF G 1 X F GF G, the following verification of X satisfies the three conditions in the group inverse definition: AXA FGF GF GFG FGFF G F G GFG FG A, XAX F GF GFGF GF G FF G F G GFGFF G F G G FF G F G G F GF G X, AX FGF GF G FGFF G F G G I, XA F GF GFG FF G F G GFG I, so X is the group inverse of A FG..

4 354 ei Wang and Junqing Wang heorem 1 mm Let the full rank decomposition of A is A FG, where F mr is column full rank decomposition, G rn ranka rankf rankg r r 0, then the group inverse of the m- order centrosymmetric matrix A JA AJ JAJ is: 1 A A J. 4 JA JA J mm Proof: Let the full rank decomposition of A is A FG, by theorem, we F JF have G GJ therefore: is the full rank decomposition of A AJ JA JAJ, 1 F F JF JF F 1 F JF 4 JF = G GJ G GJ 1 1 GF G GJ GF G GJ F GF G F GF GJ 1 A A J JF GF G JF GF GJ JA JA J heorem mm Let the full rank decomposition of A is A FG, where F mr is column full rank decomposition, G rn ranka rankf rankg r r 0, then the group inverse of the m+1- order A centrosymmetric matrix 0 JA AJ is: JAJ A A J JA JA J mm Proof: Let the full rank decomposition of A is A FG, by theorem, we

5 Group inverse for a class of centrosymmetric matrix 355 F JF have G GJ therefore: A is the full rank decomposition of 0 JA 1 F F JF JF F F 1 4 JF JF = G GJ G GJ 1 1 GF G GJ GF G GJ 1 1 F GF G F GF GJ A JF GF G JF GF GJ JA 3. Conclusion A J 0. JA J AJ, JAJ o sum up, for the full rank decomposition of the centrosymmetric matrix, it is found that the full rank decomposition of the mother matrix A can be performed first, and then the full rank decomposition can be easily calculated by using the conclusions in the literature. For the group inverse of the centrosymmetric matrix, through research, we can find the group inverse of its mother matrix and then use the theorem of this paper to calculate the result. herefore, it is very easy to calculate the group inverse of the centrosymmetric matrix through the inverse of the mother matrix. he calculation method not only reduces the amount of calculation, but also it does not lose accuracy. References [1] Changjiang Pu, Yimin Wei, he Symbolic Pattern of Generalized Inverse, Beijing: Science Press, 014. [] Wei Guo, Full Rank Decomposition and oore-penrose Inverses of Completely Symmetric atrices, Journal of Sichuan Normal University (Natural Science), 3 (009), no. 4,

6 356 ei Wang and Junqing Wang [3] Zhong Xu, Quan Lu, he Inverse and Generalized Inverses of Centrosymmetric and Anti-centrosymmetric Symmetric atrices, athematical Bulletin, 3 (1998), [4] JingPin Huang, he Drazin Inverse of a Centrosymmetric atrix, Journal of Chongqing eachers College (Natural Science Edition), 16 (1999), no. 1, Received: June 3, 018; Published: June 9, 018