complex conjugate matrix eqatrions for systems and control

Communications and Control Engineering

Ai-Guo Wu
Ying Zhang

Complex Conjugate
Matrix Equations
for Systems and
Control


Communications and Control Engineering
Series editors
Alberto Isidori, Roma, Italy
Jan H. van Schuppen, Amsterdam, The Netherlands
Eduardo D. Sontag, Piscataway, USA
Miroslav Krstic, La Jolla, USA


More information about this series at />

Ai-Guo Wu Ying Zhang
•

Complex Conjugate Matrix
Equations for Systems
and Control

123

Ai-Guo Wu
Harbin Institute of Technology, Shenzhen
University Town of Shenzhen
Shenzhen
China

Ying Zhang
Harbin Institute of Technology, Shenzhen
University Town of Shenzhen
Shenzhen
China

ISSN 0178-5354
ISSN 2197-7119 (electronic)
Communications and Control Engineering
ISBN 978-981-10-0635-7
ISBN 978-981-10-0637-1 (eBook)
DOI 10.1007/978-981-10-0637-1
Library of Congress Control Number: 2016942040
Mathematics Subject Classification (2010): 15A06, 11Cxx
© Springer Science+Business Media Singapore 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
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Printed on acid-free paper
This Springer imprint is published by Springer Nature
The registered company is Springer Science+Business Media Singapore Pte Ltd.


To our supervisor, Prof. Guang-Ren Duan
To Hong-Mei, and Yi-Tian
To Rui, and Qi-Yu

(Ai-Guo Wu)

(Ying Zhang)


Preface

Theory of matrix equations is an important branch of mathematics, and has broad
applications in many engineering fields, such as control theory, information theory,
and signal processing. Specifically, algebraic Lyapunov matrix equations play vital
roles in stability analysis for linear systems, and coupled Lyapunov matrix equations appear in the analysis for Markovian jump linear systems; algebraic Riccati
equations are encountered in optimal control. Due to these reasons, matrix equations are extensively investigated by many scholars from various fields, and the
content on matrix equations has been very rich. Matrix equations are often covered
in some books on linear algebra, matrix analysis, and numerical analysis. We list
several books here, for example, Topics in Matrix Analysis by R.A. Horn and
C.R. Johnson [143], The Theory of Matrices by P. Lancaster and M. Tismenetsky
[172], and Matrix Analysis and Applied Linear Algebra by C.D. Meyer [187]. In
addition, there are some books on special matrix equations, for example, Lyapunov

Matrix Equations in System Stability and Control by Z. Gajic [128], Matrix Riccati
Equations in Control and Systems Theory by H. Abou-Kandil [2], and Generalized
Sylvester Equations: Unified Parametric Solutions by Guang-Ren Duan [90]. It
should be pointed out that all the matrix equations investigated in the aforementioned books are in real domain. By now, it seems that there is no book on complex
matrix equations with the conjugate of unknown matrices. For convenience, this
class of equations is called complex conjugate matrix equations.
The first author of this book and his collaborators began to consider complex
matrix equations with the conjugate of unknown matrices in 2005 inspired by the
work [155] of Jiang published in Linear Algebra and Applications. Since then, he
and his collaborators have published many papers on complex conjugate matrix
equations. Recently, the second author of this book joined this field, and has
obtained some interesting results. In addition, some complex conjugate matrix
equations have found applications in the analysis and design of antilinear systems.
This book aims to provide a relatively systematic introduction to complex conjugate
matrix equations and its applications in discrete-time antilinear systems.

vii


viii

Preface

The book has 12 chapters. In Chap. 1, first a survey is given on linear matrix
equations, and then recent development on complex conjugate matrix equations is
summarized. Some mathematical preliminaries to be used in this book are collected
in Chap. 2. Besides these two chapters, the rest of this book is partitioned into three
parts. The first part contains Chaps. 3–5, and focuses on the iterative solutions for
several types of complex conjugate matrix equations. The second part consists of
Chaps. 6–10, and focuses on explicit closed-form solutions for some complex

conjugate matrix equations. In the third part, including Chaps. 11 and 12, several
applications of complex conjugate matrix equations are considered. In Chap. 11,
stability analysis of discrete-time antilinear systems is investigated, and some stability criteria are given in terms of anti-Lyapunov matrix equations, which are
special complex conjugate matrix equations. In Chap. 12, some feedback design
problems are solved for discrete-time antilinear systems by using several types of
complex conjugate matrix equations. Except part of Chap. 2 and Subsection 6.1.1,
the other materials of this book are based on our own research work, including
some unpublished results.
The intended audience of this monograph includes students and researchers in
areas of control theory, linear algebra, communication, numerical analysis, and so
on. An appropriate background for this monograph would be the first course on
linear algebra and linear systems theory.
Since 1980s, many researchers have devoted much effort in complex conjugate
matrix equations, and much contribution has been made to this area. Owing to
space limitation and the organization of the book, many of their published results
are not included or even not cited. We extend our apologies to these researchers.
It is under the supervision of our Ph.D. advisor, Prof. Guang-Ren Duan at
Harbin Institute of Technology (HIT), that we entered the field of matrix equations
with their applications in control systems design. Moreover, Prof. Duan has also
made much contribution to the investigation of complex conjugate matrix equations, and has coauthored many papers with the first author. Some results in these
papers have been included in this book. Therefore, at the beginning of preparing the
manuscript, we intended to get Prof. Duan as the first author of this book due to his
contribution on complex conjugate matrix equations. However, he thought that he
did not make contribution to the writing of this book, and thus should not be an
author of this book. Here, we wish to express our sincere gratitude and appreciation
to Prof. Duan for his magnanimity and selflessness. We also would like to express
our profound gratitude to Prof. Duan for his careful guidance, wholehearted support, insightful comments, and great contribution.
We also would like to give appreciation to our colleague, Prof. Bin Zhou of HIT
for his help. The first author has coauthored some papers included in this book with
Prof. Gang Feng when he visited City University of Hong Kong as a Research

Fellow. The first author would like to express his sincere gratitude to Prof. Feng for
his help and contribution. Dr. Yan-Ming Fu, Dr. Ming-Zhe Hou, Mr. Yang-Yang
Qian, and Dr. Ling-Ling Lv have also coauthored with the first author a few papers
included in this book. The first author would extend his great thanks to all of them
for their contribution.


Preface

ix

Great thanks also go to Mr. Yang-Yang Qian and Mr. Ming-Fang Chang, Ph.D.
students of the first author, who have helped us in typing a few sections of the
manuscripts. In addition, Mr. Fang-Zhou Fu, Miss Dan Guo, Miss Xiao-Yan He,
Mr. Zhen-Peng Zeng, and Mr. Tian-Long Qin, Master students of the first author,
and Mr. Yang-Yang Qian and Mr. Ming-Fang Chang have provided tremendous
help in finding errors and typos in the manuscripts. Their help has significantly
improved the quality of the manuscripts, and is much appreciated.
The first and second authors would like to thank his wife Ms. Hong-Mei Wang
and her husband Dr. Rui Zhang, respectively, for their constant support in every
aspect. Part of the book was written when the first author visited the University of
Western Australia (UWA) from July 2013 to July 2014. The first author would like
to thank Prof. Victor Sreeram at UWA for his help and invaluable suggestions.
We would like to gratefully acknowledge the financial support kindly provided
by the National Natural Science Foundation of China under Grant Nos.
60974044 and 61273094, by Program for New Century Excellent Talents in
University under Grant No. NCET-11-0808, by Foundation for the Author of
National Excellent Doctoral Dissertation of China under Grant No. 201342, by
Specialized Research Fund for the Doctoral Program of Higher Education under
Grant Nos. 20132302110053 and 20122302120069, by the Foundation for Creative

Research Groups of the National Natural Science Foundation of China under Grant
Nos. 61021002 and 61333003, by the National Program on Key Basic Research
Project (973 Program) under Grant No. 2012CB821205, by the Project for
Distinguished Young Scholars of the Basic Research Plan in Shenzhen City under
Contract No. JCJ201110001, and by Key Laboratory of Electronics Engineering,
College of Heilongjiang Province (Heilongjiang University).
Lastly, we thank in advance all the readers for choosing to read this book. It is
much appreciated if readers could possibly provide, via email: ,
feedback about any problems found.
July 2015

Ai-Guo Wu
Ying Zhang


Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Univariate Linear Matrix Equations . . . . . . . . . . . . . . . . .
1.2.1 Lyapunov Matrix Equations . . . . . . . . . . . . . . . .
1.2.2 Kalman-Yakubovich and Normal Sylvester Matrix
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Other Matrix Equations. . . . . . . . . . . . . . . . . . . .
1.3 Multivariate Linear Matrix Equations . . . . . . . . . . . . . . . .
1.3.1 Roth Matrix Equations . . . . . . . . . . . . . . . . . . . .

1.3.2 First-Order Generalized Sylvester Matrix Equations
1.3.3 Second-Order Generalized Sylvester Matrix
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 High-Order Generalized Sylvester Matrix
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 Linear Matrix Equations with More Than Two
Unknowns. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Coupled Linear Matrix Equations. . . . . . . . . . . . . . . . . . .
1.5 Complex Conjugate Matrix Equations. . . . . . . . . . . . . . . .
1.6 Overview of This Monograph . . . . . . . . . . . . . . . . . . . . .

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Mathematical Preliminaries . . . . . . . . . .
2.1 Kronecker Products . . . . . . . . . . . .
2.2 Leverrier Algorithms . . . . . . . . . . .
2.3 Generalized Leverrier Algorithms. . .
2.4 Singular Value Decompositions . . . .
2.5 Vector Norms and Operator Norms .
2.5.1 Vector Norms . . . . . . . . . .

2.5.2 Operator Norms . . . . . . . . .
2.6 A Real Representation of a Complex
2.6.1 Basic Properties . . . . . . . . .
2.6.2 Proof of Theorem 2.7 . . . . .

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xi


xii

Contents

2.7
2.8

Consimilarity. . . . . . . . . . . . . . . . . . . . . . . .
Real Linear Spaces and Real Linear Mappings
2.8.1 Real Linear Spaces. . . . . . . . . . . . . .
2.8.2 Real Linear Mappings . . . . . . . . . . .
2.9 Real Inner Product Spaces. . . . . . . . . . . . . . .
2.10 Optimization in Complex Domain . . . . . . . . .
2.11 Notes and References . . . . . . . . . . . . . . . . . .
Part I

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Iterative Solutions

3

Smith-Type Iterative Approaches . . . . . . . . . . .
3.1 Infinite Series Form of the Unique Solution.
3.2 Smith Iterations . . . . . . . . . . . . . . . . . . . .
3.3 Smith (l) Iterations . . . . . . . . . . . . . . . . . .
3.4 Smith Accelerative Iterations . . . . . . . . . . .
3.5 An Illustrative Example . . . . . . . . . . . . . .
3.6 Notes and References . . . . . . . . . . . . . . . .

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4

Hierarchical-Update-Based Iterative Approaches . . . . . . .
4.1 Extended Con-Sylvester Matrix Equations. . . . . . . . .
4.1.1 The Matrix Equation AXB þ CXD ¼ F . . . . .
4.1.2 A General Case . . . . . . . . . . . . . . . . . . . . .
4.1.3 Numerical Examples. . . . . . . . . . . . . . . . . .
4.2 Coupled Con-Sylvester Matrix Equations . . . . . . . . .
4.2.1 Iterative Algorithms . . . . . . . . . . . . . . . . . .
4.2.2 Convergence Analysis . . . . . . . . . . . . . . . .
4.2.3 A More General Case . . . . . . . . . . . . . . . . .
4.2.4 A Numerical Example . . . . . . . . . . . . . . . .
4.3 Complex Conjugate Matrix Equations with Transpose
of Unknowns. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Convergence Analysis . . . . . . . . . . . . . . . .
4.3.2 A Numerical Example . . . . . . . . . . . . . . . .
4.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . .

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119
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147

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149
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158

Finite Iterative Approaches . . . . . . . . . . . . . . . . . .
5.1 Generalized Con-Sylvester Matrix Equations . . .
5.1.1 Main Results . . . . . . . . . . . . . . . . . . .
5.1.2 Some Special Cases . . . . . . . . . . . . . .
5.1.3 Numerical Examples. . . . . . . . . . . . . .
5.2 Extended Con-Sylvester Matrix Equations. . . . .
5.2.1 The Matrix Equation AXB þ CXD ¼ F .
5.2.2 A General Case . . . . . . . . . . . . . . . . .
5.2.3 Numerical Examples. . . . . . . . . . . . . .

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5

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Contents

5.3

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198
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6

Real-Representation-Based Approaches. . . . . . . .
6.1 Normal Con-Sylvester Matrix Equations . . . .
6.1.1 Solvability Conditions . . . . . . . . . .
6.1.2 Uniqueness Conditions . . . . . . . . . .
6.1.3 Solutions. . . . . . . . . . . . . . . . . . . .
6.2 Con-Kalman-Yakubovich Matrix Equations . .
6.2.1 Solvability Conditions . . . . . . . . . .
6.2.2 Solutions. . . . . . . . . . . . . . . . . . . .
6.3 Con-Sylvester Matrix Equations. . . . . . . . . .
6.4 Con-Yakubovich Matrix Equations. . . . . . . .
6.5 Extended Con-Sylvester Matrix Equations. . .
6.6 Generalized Con-Sylvester Matrix Equations .
6.7 Notes and References . . . . . . . . . . . . . . . . .

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225
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7

Polynomial-Matrix-Based Approaches. . . . . . . . . . . . .
7.1 Homogeneous Con-Sylvester Matrix Equations . . .
7.2 Nonhomogeneous Con-Sylvester Matrix Equations.
7.2.1 The First Approach . . . . . . . . . . . . . . . .
7.2.2 The Second Approach . . . . . . . . . . . . . .
7.3 Con-Yakubovich Matrix Equations. . . . . . . . . . . .
7.3.1 The First Approach . . . . . . . . . . . . . . . .
7.3.2 The Second Approach . . . . . . . . . . . . . .
7.4 Extended Con-Sylvester Matrix Equations. . . . . . .
7.4.1 Basic Solutions . . . . . . . . . . . . . . . . . . .
7.4.2 Equivalent Forms . . . . . . . . . . . . . . . . . .
7.4.3 Further Discussion . . . . . . . . . . . . . . . . .
7.4.4 Illustrative Examples . . . . . . . . . . . . . . .
7.5 Generalized Con-Sylvester Matrix Equations . . . . .
7.5.1 Basic Solutions . . . . . . . . . . . . . . . . . . .
7.5.2 Equivalent Forms . . . . . . . . . . . . . . . . . .
7.5.3 Special Solutions . . . . . . . . . . . . . . . . . .
7.5.4 An Illustrative Example . . . . . . . . . . . . .
7.6 Notes and References . . . . . . . . . . . . . . . . . . . . .

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275

276
284
285
293
294
295
305
307
308
311
316
318
321
322
324
329
332
334

5.4
Part II

Coupled Con-Sylvester Matrix Equations .
5.3.1 Iterative Algorithms . . . . . . . . . .
5.3.2 Convergence Analysis . . . . . . . .
5.3.3 A More General Case . . . . . . . . .
5.3.4 Numerical Examples. . . . . . . . . .
5.3.5 Proofs of Lemmas 5.15 and 5.16 .
Notes and References . . . . . . . . . . . . . . .

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Explicit Solutions


xiv

Contents

8

Unilateral-Equation-Based Approaches . . . . . . . . . . . .
8.1 Con-Sylvester Matrix Equations. . . . . . . . . . . . . .
8.2 Con-Yakubovich Matrix Equations. . . . . . . . . . . .
8.3 Nonhomogeneous Con-Sylvester Matrix Equations.
8.4 Notes and References . . . . . . . . . . . . . . . . . . . . .

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335
336
343
349
354

9

Conjugate Products . . . . . . . . . . . . . . . . . . . . . . .
9.1 Complex Polynomial Ring ðC½sŠ; þ ; ~Þ. . . . .
9.2 Division with Remainder in ðC½sŠ; þ ; ~Þ . . . .
9.3 Greatest Common Divisors in ðC½sŠ; þ ; ~Þ . .
9.4 Coprimeness in ðC½sŠ; þ ; ~Þ . . . . . . . . . . . .

9.5 Conjugate Products of Polynomial Matrices. . .
9.6 Unimodular Matrices and Smith Normal Form.
9.7 Greatest Common Divisors . . . . . . . . . . . . . .
9.8 Coprimeness of Polynomial Matrices . . . . . . .
9.9 Conequivalence and Consimilarity . . . . . . . . .
9.10 An Example . . . . . . . . . . . . . . . . . . . . . . . .
9.11 Notes and References . . . . . . . . . . . . . . . . . .

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355
355
359
362
365
366
371
377
379
382
385
385

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389
389
394
394
397
400
402

11 Stability for Antilinear Systems . . . . . . . . . . . . . . . . . . .
11.1 Stability for Discrete-Time Antilinear Systems. . . . . .
11.2 Stochastic Stability for Markovian Antilinear Systems
11.3 Solutions to Coupled Anti-Lyapunov Equations . . . . .
11.3.1 Explicit Iterative Algorithms . . . . . . . . . . . .
11.3.2 Implicit Iterative Algorithms . . . . . . . . . . . .
11.3.3 An Illustrative Example . . . . . . . . . . . . . . .
11.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . .
11.4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.2 A Brief Overview . . . . . . . . . . . . . . . . . . .

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405
407
410
423
424
428
432
435
435
436

....
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Gain
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439
439
442
443
445
446
447

10 Con-Sylvester-Sum-Based Approaches . . . . . . .
10.1 Con-Sylvester Sum. . . . . . . . . . . . . . . . . .
10.2 Con-Sylvester-Polynomial Matrix Equations
10.2.1 Homogeneous Case . . . . . . . . . . .
10.2.2 Nonhomogeneous Case . . . . . . . . .
10.3 An Illustrative Example . . . . . . . . . . . . . .
10.4 Notes and References . . . . . . . . . . . . . . . .
Part III

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Applications in Systems and Control

12 Feedback Design for Antilinear Systems . . . . . . . . . . .
12.1 Generalized Eigenstructure Assignment. . . . . . . . .
12.2 Model Reference Tracking Control. . . . . . . . . . . .
12.2.1 Tracking Conditions . . . . . . . . . . . . . . . .
12.2.2 Solution to the Feedback Stabilizing Gain .
12.2.3 Solution to the Feedforward Compensation
12.2.4 An Example . . . . . . . . . . . . . . . . . . . . .


Contents

12.3 Finite Horizon Quadratic Regulation. .
12.4 Infinite Horizon Quadratic Regulation.
12.5 Notes and References . . . . . . . . . . . .
12.5.1 Summary . . . . . . . . . . . . . .
12.5.2 A Brief Overview . . . . . . . .

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450
461
467
467
468

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Notation

Notation Related to Subspaces
Z
R
C
Rn
Cn
RmÂn
CmÂn
RmÂn ½sŠ
CmÂn ½sŠ
Ker
Image
rdim

Set of all integer numbers
Set of all real numbers
Set of all complex numbers
Set of all real vectors of dimension n
Set of all complex vectors of dimension n
Set of all real matrices of dimension m  n
Set of all complex matrices of dimension m  n
Set of all polynomial matrices of dimension m  n with real
coefficients
Set of all polynomial matrices of dimension m  n with complex
coefficients
The kernel of a mapping
The image of a mapping
The real dimension of a real linear space

Notation Related to Vectors and Matrices
0n
0mÂn
IÂn Ã
aij mÂn
AÀ1
AT
A
AH
n

diag Ai

Zero vector in Rn
Zero matrix in RmÂn
Identity matrix of order n
Matrix of dimension m  n with the i-th row and j-th column element
being aij
Inverse matrix of matrix A
Transpose of matrix A
Complex conjugate of matrix A
Transposed complex conjugate of matrix A
The matrix whose i-th block diagonal is Ai

i¼1

xvii

xviii

Notation

ReðAÞ
ImðAÞ
detðAÞ
adjðAÞ
trðAÞ
rankðAÞ
vecðAÞ

qð AÞ
kð AÞ
kmin ð AÞ
kmax ð AÞ
rmax ð AÞ
kAk2
kAk

Real part of matrix A
Imaginary part of matrix A
Determinant of matrix A
Adjoint of matrix A
Trace of matrix A
Rank of matrix A
Vectorization of matrix A
Kronecker product of two matrices
Spectral radius of matrix A
Set of the eigenvalues of matrix A

The minimal eigenvalue of matrix A
The maximal eigenvalue of matrix A
The maximal singular value of matrix A
2-norm of matrix A
Frobenius norm of matrix A

Ak

The k-th right alternating power of matrix A

Ak
kðE; AÞ

The k-th left alternating power of matrix A
Set of the finite eigenvalues of matrix pair ðE; AÞ

!

Other Notation
E
i
I½m; nŠ
min
max
~

Mathematical expectation
The imaginary unit
The set of integers from m to n
The minimum value in a set

The maximum value in a set
Conjugate product of two polynomial matrices

F

Á
()
£

Con-Sylvester sum
If and only if
Empty set


Chapter 1

Introduction

The theory of matrix equations is an active research topic in matrix algebra, and has
been extensively investigated by many researchers. Different matrix equations have
wide applications in various areas, such as, communication, signal processing and
control theory. Specifically, Lyapunov matrix equations are often encountered in stability analysis of linear systems [160]; the homogeneous continuous-time Lyapunov
equation in block companion matrices plays a vital role in the investigation of factorizations of Hermitian block Hankel matrices [228]; generalized Sylvester matrix
equations are often encountered in eigenstructure assignment of linear systems [90].
As to a matrix equation, three basic problems need to be considered: the solvability conditions, solving approaches and expressions of the solutions. For real matrix
equations, a considerable number of results have been obtained for these problems.
In addition, some other problems have also been considered for some special matrix
equations. For example, geometric properties of continuous-time Lyapunov matrix
equations were investigated in [286]; bounds of the solution were studied for discretetime algebraic Lyapunov equations in [173, 227] and for continuous-time Lyapunov
equations in [173]. However, there are only a few results on complex matrix equations with the conjugate of unknown matrices reported in literature. For convenience,

the type of these matrix equations is called the complex conjugate matrix equation.
Recently, complex conjugate matrix equations have found some applications in
discrete-time antilinear systems. In this book, some recent results are summarized
for several kinds of complex conjugate matrix equations and their applications in
analysis and feedback design of antilinear systems. In this chapter, the main aim is to
first provide a survey on real linear matrix equations, and then give recent progress
on complex conjugate matrix equations. The recent progress on antilinear systems
and related problems will be given in the part of “Notes and References” of Chaps. 11
and 12. At the end of this chapter, an overview of this monograph is presented.
Symbols used in this chapter are now introduced. It should be pointed out that these
symbols are also adopted throughout this book. For two integers m ≤ n, the notation
© Springer Science+Business Media Singapore 2017
A.-G. Wu and Y. Zhang, Complex Conjugate Matrix Equations
for Systems and Control, Communications and Control Engineering,
DOI 10.1007/978-981-10-0637-1_1

1


2

1 Introduction

I[m, n] denotes the set {m, m + 1, . . . , n}. For a square matrix A, we use det A,
ρ (A), λ (A), λmin (A) , and λmax (A) to denote the determinant, the spectral radius,
the set of eigenvalues, the minimal and maximal eigenvalues of A, respectively. The
notations A, AT , and AH denote the conjugate, transpose and conjugate transpose of
the matrix A, respectively. Re (A) and Im (A) denote the real part and imaginary part
n

of the matrix A, respectively. In addition, diag Ai is used to denote the block diagonal
i=1

matrix whose elements in the main block-diagonal are Ai , i ∈ I[1, n]. The symbol
“⊗” is used to denote the Kronecker product of two matrices.

1.1 Linear Equations
The most common linear equation may be the following real equation
Ax = b,

(1.1)

where A ∈ Rm×n and b ∈ Rm are known, and x ∈ Rn is the vector to be determined.
If A is a square matrix, it is well-known that the linear equation (1.1) has a unique
solution if and only if the matrix A is invertible, and in this case, the unique solution
can be given by x = A−1 b. In addition, this unique solution can also be given by
xi =

det Ai
, i ∈ I[1, n],
det A

where xi is the i-th element of the vector x, and Ai is the matrix formed by replacing
the i-th column of A with the column vector b. This is the celebrated Cramer’s rule.
For the general case, it is well-known that the matrix equation (1.1) has a solution if
and only if
rank A b = rankA.
In addition, the solvability of the general equation (1.1) can be characterized in terms
of generalized inverses, and the general expression of all the solutions to the equation
(1.1) can also be given in terms of generalized inverses.

Definition 1.1 ([206, 208]) Given a matrix A ∈ Rm×n , if a matrix X ∈ Rn×m satisfies
AXA = A,
then X is called a generalized inverse of the matrix A.
The generalized inverse may be not unique. An arbitrary generalized inverse of
the matrix A is denoted by A− .
Theorem 1.1 ([208, 297]) Given a matrix A ∈ Rm×n , let A− be an arbitrary generalized inverse of A. Then, the vector equation (1.1) has a solution if and only
if


1.1 Linear Equations

3

AA− b = b.

(1.2)

Moreover, if the condition (1.2) holds, then all the solutions of the vector equation
(1.1) can be given by
x = A− b + I − A− A z,
where z is an arbitrary n-dimensional vector.
The analytical solutions of the equation (1.1) given by inverses or generalized
inverses have neat expressions, and play important roles in theoretical analysis. However, it has been recognized that the operation of matrix inverses is not numerically
reliable. Therefore, many numerical methods are applied in practice to solve linear
vector equations. These methods can be classified into two types. One is the transformation approach, in which the matrix A needs to be transformed into some special
canonical forms, and the other is the iterative approach which generates a sequence
of vectors that approach the exact solution. An iterative process may be stopped as
soon as an approximate solution is sufficiently accurate in practice.
For the equation (1.1) with m = n, the celebrated iterative methods include Jacobi
iteration and Gauss-Seidel iteration. Let

⎡ ⎤
⎡ ⎤
x1
b1
⎢ x2 ⎥
⎢ b2 ⎥
⎥
⎢
⎢
.
A = aij n×n , b = ⎣ ⎦ , x = ⎣ ⎥
···
···⎦
bn
xn
Then, the vector equation (1.1) can be explicitly written as
⎧
a11 x1 + a12 x2 + · · · + a1n xn = b1
⎪
⎪
⎨
a21 x1 + a22 x2 + · · · + a2n xn = b2
.
···
⎪
⎪
⎩
an1 x1 + an2 x2 + · · · + ann xn = bn
The Gauss-Seidel and Jacobi iterative methods require that the vector equation (1.1)
has a unique solution, and all the entries in the main diagonal of A are nonzero, that

is, aii = 0, i ∈ I[1, n]. It is assumed that the initial values xi (0) of xi , i ∈ I[1, n],
are given. Then, the Jacobi iterative method obtains the unique solution of (1.1) by
the following iteration [132]:
⎛
1 ⎝
xi (k + 1) =
bi −
aii

i−1

n

aij xj (k) −
j=1

⎞
aij xj (k)⎠ , i ∈ I[1, n],

j=i+1

and the Gauss-Seidel iterative method obtains the unique solution of (1.1) by the
following forward substitution [132]:


4

1 Introduction

⎛

1 ⎝
bi −
xi (k + 1) =
aii

i−1

⎞

n

aij xj (k)⎠ , i ∈ I[1, n].

aij xj (k + 1) −
j=1

j=i+1

Now, let
n

D = diag aii ,
i=1
⎡
0 0
⎢ a21 0
⎢
L=⎢
⎢ a31 a32
⎣··· ···

an1 an2

0 ···
0 ···
0 ···
··· ···
· · · an,n−1

⎡
⎤
0
0
⎢ 0
0 ⎥
⎢
⎥
⎢
0 ⎥
⎥, U = ⎢ 0
⎣···
⎦
···
0
0

a12
0
0
···
0

a13
a23
0
···
0

⎤
· · · a1n
· · · a2n ⎥
⎥
··· ··· ⎥
⎥.
· · · an−1,n ⎦
··· 0

Then, the Gauss-Seidel iterative algorithm can be written in the following compact
form:
x (k + 1) = − (D + L)−1 Ux (k) + (D + L)−1 b,
and the Jacobi iterative algorithm can be written as
x (k + 1) = −D−1 (L + U) x (k) + D−1 b.
It is easily known that the Gauss-Seidel iteration is convergent if and only if
ρ (D + L)−1 U < 1,
and the Jacobi iteration is convergent if and only if
ρ D−1 (L + U) < 1.
In general, the Gauss-Seidel iteration converges faster than the Jacobi iteration since
the recent estimation is used in the Gauss-Seidel iteration. However, there exist
examples where the Jacobi method is faster than the Gauss–Seidel method.
In 1950, David M. Young, Jr. and H. Frankel proposed a variant of the GaussSeidel iterative method for solving the equation (1.1) with m = n [156]. This is
the so-called successive over-relaxation (SOR) method, by which the elements xi ,

i ∈ I[1, n], of x can be computed sequentially by forward substitution:
⎛
i−1
ω ⎝
xi (k + 1) = (1 − ω) xi (k) +
aij xj (k + 1) −
bi −
aii
j=1

n

⎞
aij xj (k)⎠ , i ∈ I[1, n],

j=i+1

where the constant ω > 1 is called the relaxation factor. Analytically, this algorithm
can be written as


1.1 Linear Equations

5

x (k + 1) = (D + ωL)−1 [ωb − (ωU + (ω − 1) D) x (k)] .
The choice of relaxation factor is not necessarily easy, and depends on the properties
of A. It has been proven that if A is symmetric and positive definite, the SOR method
is convergent with 0 < ω < 2.
If A is symmetric and positive definite, the equation (1.1) can be solved by the

conjugate gradient method proposed by Hestenes and Stiefel. This method is given
in the following theorem.
Theorem 1.2 Given a symmetric and positive definite matrix A ∈ Rn×n , the solution
of the equation (1.1) can be obtained by the following iteration
⎧
T
(k)p(k)
⎪
α (k) = prT (k)Ap(k)
⎪
⎪
⎪
⎪
⎨ x (k + 1) = x (k) + α (k) p (k)
r (k + 1) = b − Ax (k + 1)
,
⎪
pT (k)Ar(k+1)
⎪
⎪
s (k) = − pT (k)Ap(k)
⎪
⎪
⎩
p (k + 1) = r (k + 1) + s (k) p (k)
where the initial conditions x (0), r (0) , and p (0) are given as x (0) = x0 ,
and p (0) = r (0) = b − Ax (0).

1.2 Univariate Linear Matrix Equations
In this section, a simple survey is provided for linear matrix equations with only one

unknown matrix variable. Let us start with the Lyapunov matrix equations.

1.2.1 Lyapunov Matrix Equations
The most celebrated univariate matrix equations may be the continuous-time and
discrete-time Lyapunov matrix equations, which play vital roles in stability analysis [75, 160], controllability and observability analysis of linear systems [3]. The
continuous-time and discrete-time Lyapunov matrix equations are respectively in
the forms as
AT X + XA = −Q,

(1.3)

X − A XA = Q,

(1.4)

T

where A ∈ Rn×n , and positive semidefinite matrix Q ∈ Rn×n are known, and X is the
matrix to be determined. In [103, 104], the robust stability analysis was investigated
for linear continuous-time and discrete-time systems, respectively, and the admissible perturbation bounds of the system matrices were given in terms of the solutions


6

1 Introduction

of the corresponding Lyapunov matrix equations. In [102], the robust stability was
considered for linear continuous-time systems subject to unmodeled dynamics, and
an admissible bound was given for the nonlinear perturbation function based on the
solution to the Lyapunov matrix equation of the nominal linear system. In linear

systems theory, it is well-known that the controllability and observability of a linear system can be checked by the existence of a positive definite solution to the
corresponding Lyapunov matrix equation [117].
In [145], the continuous-time Lyapunov matrix equation was used to analyze the
weighted logarithmic norm of matrices. While in [106], this equation was employed
to investigate the so-called generalized positive definite matrix. In [222], the inverse
solution of the discrete-time Lyapunov equation was applied to generate q-Markov
covers for single-input-single-output discrete-time systems. In [317], a relationship
between the weighted norm of a matrix and the corresponding discrete-time Lyapunov matrix equation was first established, and then an iterative algorithm was
presented to obtain the spectral radius of a matrix by the solutions of a sequence of
discrete-time Lyapunov matrix equations.
For the solutions of Lyapunov matrix equations with special forms, many results
have been reported in literature. When A is in the Schwarz form, and Q is in a special
diagonal form, the solution of the continuous-time Lyapunov matrix equation (1.3)
was explicitly given in [12]. When AT is in the following companion form:
⎡

⎤
0 1
⎢ ..
⎥
.. . .
⎢
⎥
.
.
AT = ⎢ .
⎥,
⎣ 0 0 ··· 1 ⎦
−a0 −a1 · · · −an−1
T

and Q = bbT with b = 0 0 · · · 1 , it was shown in [221] that the solution of the
Lyapunov matrix equation (1.3) with A Hurwitz stable can be given by using the
entries of a Routh table. In [165], a simple algorithm was proposed for a closedform solution to the continuous-time Lyapunov matrix equation (1.3) by using the
Routh array when A is in a companion form. In [19], the solutions for the above two
Lyapunov matrix equations, which are particularly suitable for symbolic implementation, were proposed for the case where the matrix A is in a companion form. In
[24], the following special discrete-time Lyapunov matrix equation was considered:
X − FXF T = GQG T ,
where the matrix pair (F, G) is in a controllable canonical form. It was shown in [24]
that the solution to this equation is the inverse of a Schur-Cohn matrix associated
with the characteristic polynomial of F.
When A is Hurwitz stable, the unique solution to the continuous-time Lyapunov
matrix equation (1.3) can be given by the following integration form [28]:


1.2 Univariate Linear Matrix Equations

7
∞

X=

T

eA t QeAt d t.

(1.5)

0

Further, let Q = BBT with B ∈ Rn×r , and let the matrix exponential function eAt be
expressed as a finite sum of the power of A:
eAt = a1 (t) I + a2 (t) A + · · · + an (t) An−1 .
Then, it was shown in [193] that the unique solution of (1.3) can also be expressed
by
(1.6)
X = Ctr (A, B) H Ctr T (A, B)
where
Ctr (A, B) = B AB · · · An−1 B
is the controllability matrix of the matrix pair (A, B), and H = G ⊗ Ir with G =
G T = gij n×n ,
∞

gij =

ai (t) aj (t) d t.

0

The expression in (1.6) may bring much convenience for the analysis of linear systems
due to the appearance of the controllability matrix. In addition, with the help of the
expression (1.6) some eigenvalue bounds of the solution to the equation (1.3) were
given in [193]. In [217], an infinite series representation of the unique solution to
the continuous-time Lyapunov matrix equation (1.3) was also given by converting it
into a discrete-time Lyapunov matrix equation.
When A is Schur stable, the following theorem summarizes some important properties of the discrete-time Lyapunov matrix equation (1.4).
Theorem 1.3 ([212]) If A is Schur stable, then the solution of the discrete-time
Lyapunov matrix equation (1.4) exists for any matrix Q, and is given as
X=

1
2π

2π

AT − ei θ I

−1

Q A − e− i θ I

−1

d θ,

0

or equivalently by

∞

X=

i

AT QAi .
i=0

Many numerical algorithms have been proposed to solve the Lyapunov matrix
equations. In view that the solution of the Lyapunov matrix equation (1.3) is at least

semidefinite, Hammarling [136] found an ingenuous way to compute the Cholesky
factor of X directly. The basic idea is to apply triangular structure to solve the
equation iteratively. By constructing a new rank-1 updating scheme, an improved
Hammarling method was proposed in [220] to accommodate a more general case


8

1 Introduction

of Lyapunov matrix equations. In [284], by using a dimension-reduced method an
algorithm was proposed to solve the continuous-time Lyapunov matrix equation
(1.3) in controllable canonical forms. In [18], the presented Smith iteration for the
discrete-time Lyapunov matrix equation (1.4) was in the form of
X (k + 1) = AT X (k) A + Q
with X (0) = Q.
Besides the solutions to Lyapunov matrix equations, the bounds of the solutions
have also been extensively investigated. In [191], the following result was given on
the eigenvalue bounds of the discrete-time Lyapunov matrix equation
X − AXAT = BBT ,

(1.7)

where A ∈ Rn×n and B ∈ Rn×r are known matrices, and X is the matrix to be
determined.
Theorem 1.4 Given matrices A ∈ Rn×n and B ∈ Rn×r , for the solution X to the
discrete-time Lyapunov matrix equation (1.7) there holds
λmin Ctr (A, B) Ctr T (A, B) P ≤ X ≤ λmax Ctr (A, B) Ctr T (A, B) P,
where P is the solution to the Lyapunov matrix equation
P − An P (An )T = I.

In [227], upper bounds for the norms and trace of the solution to the discrete-time
Lyapunov matrix equation (1.7) were presented in terms of the resolvent of A. In
[124], lower and upper bounds for the trace of the solution to the continuous-time
Lyapunov matrix equation (1.3) were given in terms of the logarithmic norm of A.
In [116], lower bounds were established for the minimal and maximal eigenvalues
of the solution to the discrete-time Lyapunov equation (1.7).
Recently, parametric Lyapunov matrix equations were extensively investigated.
In [307, 315], some properties of the continuous-time parametric Lyapunov matrix
equations were given. In [307], the solution of the parametric Lyapunov equation
was applied to semiglobal stabilization for continuous-time linear systems subject to
actuator saturation; while in [315] the solution was used to design a state feedback
stabilizing law for linear systems with input delay. The discrete-time parametric Lyapunov matrix equations were investigated in [313, 314], and some elegant properties
were established.


1.2 Univariate Linear Matrix Equations

9

1.2.2 Kalman-Yakubovich and Normal Sylvester Matrix
Equations
A general form of the continuous-time Lyapunov matrix equation is the so-called
normal Sylvester matrix equation
AX − XB = C.

(1.8)

A general form of the discrete-time Lyapunov matrix equation is the so-called
Kalman-Yakubovich matrix equation
X − AXB = C.

(1.9)

In the matrix equations (1.8) and (1.9), A ∈ Rn×n , B ∈ Rp×p , and C ∈ Rn×p are the
known matrices, and X ∈ Rn×p is the matrix to be determined. On the solvability of
the normal Sylvester matrix equation (1.8), there exists the following result which
has been well-known as Roth’s removal rule.
Theorem 1.5 ([210]) Given matrices A ∈ Rn×n , B ∈ Rp×p , and C ∈ Rn×p , the
normal Sylvester matrix equation (1.8) has a solution if and only if the following two
partitioned matrices are similar
AC
,
0 B

A0
;
0B

or equivalently, the following two polynomial matrices are equivalent:
sI − A −C
,
0 sI − B

sI − A 0
.
0 sI − B

This matrix equation has a unique solution if and only if λ (A) ∩ λ (B) = ∅.
The result in the preceding theorem was generalized to the Kalman-Yakubovich
matrix equation (1.9) in [238]. This is the following theorem.

Theorem 1.6 Given matrices A ∈ Rn×n , B ∈ Rp×p , and C ∈ Rn×p , the KalmanYakubovich matrix equation (1.9) has a solution if and only if there exist nonsingular
real matrices S and R such that
S

sI + A C
sI + A 0
R=
.
0 sB + I
0 sB + I

On the numerical solutions to the normal Sylvester matrix equations, there have
been a number of results reported in literature over the past 30 years. The BartelsStewart method [13] may be the first numerically stable approach to systematically
solving small-to-medium scale Lyapunov and normal Sylvester matrix equations.
The basic idea of this method is to apply the Schur decomposition to transform the