[1] Wang Long, Chen Jianlong,. A note on the Moore-Penrose inverse of a companion matrix [J]. Journal of Southeast University (English Edition), 2017, 33 (1): 123-126. [doi:10.3969/j.issn.1003-7985.2017.01.020]

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A note on the Moore-Penrose inverse of a companion matrix()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

33

Issue:

2017 1

Page:

123-126

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2017-03-30

Info

Title:

A note on the Moore-Penrose inverse of a companion matrix

Author(s):

Wang Long; Chen Jianlong

Department of Mathematics, Southeast University, Nanjing 211189, China

Keywords:

companion matrix;  Moore-Penrose inverse;  ring

PACS:

O151.2

DOI:

10.3969/j.issn.1003-7985.2017.01.020

Abstract:

Let R be an associative ring with unity 1. The existence of the Moore-Penrose inverses of block matrices over R is investigated and the sufficient and necessary conditions for such existence are obtained. Furthermore, the representation of the Moore-Penrose inverse of M=［0 AC B］ is given under the condition of EBF=0, where E=I-CC^† and F=I-A^†A. This result generalizes the representation of the Moore-Penrose inverse of the companion matrix M=［0 aI_nn b］ due to Pedro Patrício. As for applications, some examples are given to illustrate the obtained results.

References:

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[2] Cline R E. Representations for the generalized inverse of sums of matrices [J]. Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, 1965, 2(1): 99-114. DOI:10.1137/0702008.
[3] Hung C H, Markham T L. The Moore-Penrose inverse of a partitioned matrix M=[A 0
　　B C] [J]. Czechoslovak Mathematical Journal, 1975, 25(3):354-361.
[4] Hung C H, Markham T L. The Moore-Penrose inverse of a partitioned matrix M=[A D
B C] [J]. Linear Algebra and Its Applications, 1975, 11(1):73-86. DOI:10.1016/0024-3795(75)90118-4.
[5] Hartwig R E,Patrício P. When does the Moore-Penrose inverse flip [J]. Operators and Matrices, 2012, 6(1): 181-192. DOI:10.7153/oam-06-13.
[6] Zhu H H, Chen J L, Zhang X X, et al. The Moore-Penrose inverse of 2×2 matrices over a certain *-regular ring [J]. Applied Mathematics and Computation, 2014, 246: 263-267. DOI:10.1016/j.amc.2014.08.026.
[7] Patrício P. The Moore-Penrose inverse of a companion matrix [J]. Linear Algebra and Its Applications, 2012, 437(3): 870-877. DOI:10.1016/j.laa.2012.03.019.
[8] Patrício P. The regular sum [J]. Linear and Multilinear Algebra, 2014, 63(1): 185-200. DOI:10.1080/03081087.2013.860592.
[9] Patrício P, Puystjens R. About the von Neumann regularity of triangular block matrices [J]. Linear Algebra and Its Applications, 2001, 332-334: 485-502. DOI:10.1016/s0024-3795(01)00295-6.

Memo

Memo:

Biographies: Wang Long(1988—), male, graduate; Chen Jianlong(corresponding author), male, doctor, professor, jlchen @seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.11371089), the Natural Science Foundation of Jiangsu Province(No.BK20141327), Specialized Research Fund for the Doctoral Program of Higher Education(No.20120092110020), the Natural Science Foundation of Jiangsu Higher Education Institutions of China(No.15KJB110021).
Citation: Wang Long, Chen Jianlong. A note on the Moore-Penrose inverse of a companion matrix[J].Journal of Southeast University(English Edition),2017,33(1):123-126.DOI:10.3969/j.issn.1003-7985.2017.01.020.

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