Sign Pattern for Generalized Inverses

Yimin Wei, Lizhu SUN, Changjiang BU

Description

This book addresses recent developments in sign patterns for generalized inverses. The fundamental importance of the fields is obvious, since they are related with qualitative analysis of linear systems and combinatorial matrix theory. The book provides both introductory materials and discussions to the areas in sign patterns for Moore–Penrose inverse, Drazin inverse and tensors. It is intended to convey results to the senior students and readers in pure and applied linear algebra, and combinatorial matrix theory. 

Sujets

• Sciences & Techniques
• Mathématiques
• Sciences formelles
• Mathématiques générales
• Algèbre
• Mathématiques générales
• Sciences et techniques
• Algebra
• Drazin inverse
• Linear
• Mathematics

Informations

• Publié par: EDP Sciences
• Date de parution: 10 juin 2021
• Nombre de lectures: 0
• EAN13: 9782759826001
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Current Natural Sciences

Changjiang BU, Lizhu SUN and Yimin WEI

GENERALIZED
INVERSE

Sign Pattern for Generalized
Inverses

GENERALIZED
INVERSE

ISBN : 978-2-7598-2599-8

9 782759 825998

Current Natural Sciences

Sign Pattern for Generalized
Inverses

Changjiang BU, Lizhu SUN and Yimin WEI

This book addresses recent developments in sign patterns for
generalized inverses. The fundamental importance of the fields
is obvious, since they are related with qualitative analysis of
linear systems and combinatorial matrix theory.

The book provides both introductory materials and discussions
to the areas in sign patterns for Moore–Penrose inverse, Drazin
inverse and tensors.

It is intended to convey results to the senior students and
readers in pure and applied linear algebra, and combinatorial matrix
theory.

Changjiang BU isa Professor at the College of Mathematical
Sciences, Harbin Engineering University, who works on the
graph theory and generalized inverses. He is the author of more
than 100 papers in the international journals and one monograph.

Lizhu SUN isan Associate Professor at the College of
Mathematical Sciences, Harbin Engineering University, who
works on the graph theory and multilinear algebra. She is the
author of 25 research papers.

Yimin WEIis a Professor at the School of Mathematical Sciences,
Fudan University, who works on the numerical linear algebra
and multilinear algebra. He is the author of more than 150 papers
in the international journals and six monographs published by
Science Press, Elsevier, Springer and World Scientific., etc.

www.edpsciences.org

Current Natural Sciences

Changjiang BU, Lizhu SUN
and Yimin WEI

Sign Pattern
for Generalized Inverses

Printed in France

EDP Sciences–ISBN(print): 978-2-7598-2599-8–ISBN(ebook): 978-2-7598-2600-1
DOI: 10.1051/978-2-7598-2599-8

All rights relative to translation, adaptation and reproduction by any means whatsoever
are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41
of the French Act dated March 11, 1957,“copies or reproductions reserved strictly for
private use and not intended for collective use”and, on the other hand, analyses and
short quotations for example or illustrative purposes, are allowed. Otherwise,“any
representation or reproduction–whether in full or in part–without the consent of the
author or of his successors or assigns, is unlawful”(Article 40, paragraph 1). Any
representation or reproduction, by any means whatsoever, will therefore be deemed an
infringement of copyright punishable under Articles 425 and following of the French
Penal Code.

The printed edition is not for sale in Chinese mainland. Customers in Chinese mainland
please order the print book from Science Press. ISBN of the China edition: Science Press
978-7-03-068568-1

Science Press, EDP Sciences, 2021

Preface

Generalized inverse has wide applications in science and engineering. The sign
pattern of generalized inverses has an important theoretical significance and
application value in the problems of qualitative analysis of systems and
Combinatorial Matrix Theory.
P.A. Samuelson, a Nobel Prize winner in economics, attributed the qualitative
analysis of an economics model to the problem of the sign-solvability for a linear
system, which opened the research of the sign pattern of matrices. After that,
R.A. Brualdi, an AMS fellow, and B.L. Shader, a SIAM fellow, systematically
investigated the sign pattern for the inverse of a matrix, which established the basic
theory of the sign pattern of generalized inverses. From 1995 to 2003, B.L. Shader,
Jia-Yu Shao and other scholars studied the sign pattern of the Moose–Penrose
inverse of matrices, and obtained a series of important results.
In 2010, P. van den Driessche, M. Catral and D.D. Olesky provided the group
inverses for the adjacency matrices of a class of broom graphs, and proved the sign
pattern of this kind of matrices is unique. In 2011, we proposed the concept of the
sign group invertible matrix, the strongly sign group invertible matrix and the
matrix with unique sign Drazin inverse, and established many new results on the
associated characterization. Furthermore, we extended the research to the ray
pattern of complex matrices, and characterized the matrices with ray
Moore–Penrose inverse and ray Drazin inverse. Also, we first studied the sign pattern of tensors.
The sign pattern of generalized inverses is not only an extension and a
development of the sign pattern of matrices but also a new front topic. There are many
groundbreaking research problems which need new ideas to be solved. This
monograph introduces the technical methods of characterizing the sign of generalized
inverses and review the status of the newest research on this topic, which can be used
as a reference or textbook for researchers or graduate students in matrix theory,
combinatorics algebra etc.

DOI: 10.1051/978-2-7598-2599-8.c901
© Science Press, EDP Sciences, 2021

IV

Preface

The authors are grateful to their supervisors Professors Chongguang Cao,
Baodong Zheng and Zhihao Cao for their strong support and long term guidance.
They thank Professors Jia-Yu Shao, Jipu Ma, Jiguang Sun, Erxiong Jiang, Jiaoxun
Kuang, Hongke Du, Boyin Wang, Gong-ning Chen, Hong You, Guorong Wang,
Yonglin Chen, Musheng Wei, Guoliang Chen, Wenyu Sun, Shufang Xu, Zhongxiao
Jia, Xinguo Liu, Liping Huang, Anping Liao, Jianzhou Liu, Yongzhong Song,
Hua Dai, Tingzhu Huang, Wen Li, Chuanlong Wang, Xingzhi Zhan, Yifeng Xue, An
Chang, Xiying Yuan, Chunyuan Deng, Jianlong Chen, Zhengke Miao, Xiaodong
Zhang, Yongjian Hu, Zhenghong Yang, Dongxiu Xie, Yuwen Wang, Qingwen Wang,
Qingxiang Xu, Bin Zheng, Xiaoji Liu, Hanyu Li, Xian Zhang, Xiaomin Tang,
Erfang Shan, Haiying Shan, Qianglian Huang, Haifeng Ma, Lihua You, Zhaoliang
Xu, A. Ben-Isreal, N. Castro González, M. Catral, D. CvetkovićIlić, D. Djordjević,
R. Hartwig, J. Koliha, C. Meyer, M. Nashed, P. Stanimirović, N. Thome, V.
Rakocević, Liqun Qi, Sanzheng Qiao, P. Wedin, H. Werner, Zi-cai Li, Chi-Kwong Li,
Fuzhen Zhang, Jiu Ding, Rencang Li, Zhongshan Li, Xiaoqing Jin, Li Qiu, Xiezhang
Li, Jun Ji, Xuzhou Chen, Jianming Miao, Yongge Tian and others. The authors
would like to thank Professor Eric King-wah Chu who read this book carefully and
provided valuable comments and suggestions.
This work is supported by National Natural Science Foundation of China under
grants 11371109, 11771099, 11801115, 12071097 and 12042103, Natural Science
Foundation of Heilong Jiang Province under grant QC2018002, Fundamental
Research Funds for the Central University, the Innovation Program of Shanghai
Municipal Education Committee, Shanghai Key Laboratory of Contemporary
Applied Mathematics and Key Laboratory of Mathematics for Nonlinear Sciences of
Fudan University.

1
Notations

N
½n
I
In
D
A
p
A
#
A
ð1Þ
A
þ
A
X
A

A
>
A
trðAÞ
rankðAÞ
qðAÞ
indðAÞ
RðAÞ
N ðAÞ
QðAÞ
A½a;b

A½:;b

A½a;:

positive integer set
½n ¼ f1;2;. . .;ng,n2N
unity matrix
nnunity matrix
Drazin inverse of matrixA
pD
A¼IAA
group inverse of matrixA
f1g-inverse of matrixA
Moore–Penrose inverse of matrixA
Xþ
A¼IAA
conjugate transpose of matrixA
transpose of matrixA
trace of square matrixA
rank of matrixA
term rank of matrixA
Drazin index of matrixA
range of matrixA
null space of matrixA
sign pattern class of matrixA
submatrix of matrixAwhose row index set isaand column index set
isb
submatrix of matrixAdetermined by the rows ofAand the columns
whose index is in index setb
submatrix of matrixAdetermined by the columns ofAand the rows
whose index is in index seta

1
Unless otherwise stated, the following notations will be used in the book.

VI

Aða;bÞ

ðAÞ
ij
detðAÞ
sgnðAÞ
rayðAÞ
mn
F
C
R
K
n
F
SnðmÞ
Nðq;TÞ

NrðAÞ
NcðAÞ
n1n2nk
F
½m;n
F
k
Rk
A
Lk
A
AfRkg
AfLkg

Notations

submatrix of matrixAobtained by removing the rows inaand the
columns inb
ði;jÞ-entry of matrixA
determinant of matrixA
sign pattern of matrixA
ray pattern of matrixA
set ofmnmatrices over fieldsF
complex fields
real fields
skew fields
set ofn-dimension vectors over fieldsF
set of all subsets of½mwithnelements
number of elements in finite integers setTwhich are less than or equal
toq2T
number of rows of matrixA
number of columns of matrixA
set ofn1n2 nk-dimensionk-order tensors over fieldsF
set ofmn n-dimensionk-order tensors over fieldsF
right inverse of tensorAwithk-order
left inverse of tensorAwithk-order
set of right inverse for tensorAwithk-order
set of left inverse for tensorAwithk-order

Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notations. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 1
Generalized Inverses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 MatrixDecompositions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Moore–Penrose Inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 DrazinInverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 GroupInverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 GeneralizedInverses and System of Linear Equations. . . . . . . . . . . . .
1.6 Graphand Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2
Generalized Inverses of Partitioned Matrices. . . . . . . . . . . . . . . . . . . . . . . . .
2.1 DrazinInverse of Partitioned Matrices. . . . . . . . . . . . . . . . . . . . . . . .
2.2 GroupInverse of Partitioned Matrices. . . . . . . . . . . . . . . . . . . . . . . . .
2.3 AdditiveFormulas for Drazin Inverse and Group Inverse. . . . . . . . . . .
2.4 DrazinInverse Index for Partitioned Matrices. . . . . . . . . . . . . . . . . . .

CHAPTER 3
2
SN