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Complex-Valued Matrix Derivatives

In this complete introduction to the theory of finding derivatives of scalar-, vector-,and matrix-valued functions in relation to complex matrix variables, Hjørungnesdescribes an essential set of mathematical tools for solving research problems whereunknown parameters are contained in complex-valued matrices. Self-contained and easyto follow, this singular reference uses numerous practical examples from signal process-ing and communications to demonstrate how these tools can be used to analyze andoptimize the performance of engineering systems. This is the first book on complex-valued matrix derivatives from an engineering perspective. It covers both unpatternedand patterned matrices, uses the latest research examples to illustrate concepts, andincludes applications in a range of areas, such as wireless communications, control the-ory, adaptive filtering, resource management, and digital signal processing. The bookincludes eighty-one end-of-chapter exercises and a complete solutions manual (availableon the Web).

Are Hjørungnes is a Professor in the Faculty of Mathematics and Natural Sciences atthe University of Oslo, Norway. He is an Editor of the IEEE Transactions on WirelessCommunications, and has served as a Guest Editor of the IEEE Journal of Selected Topicsin Signal Processing and the IEEE Journal on Selected Areas in Communications.

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Cambridge University Press978-0-521-19264-4 - Complex-Valued Matrix Derivatives: With Applications in Signal Processing andCommunicationsAre HjorungnesFrontmatterMore information

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This book addresses the problem of complex-valued derivatives in a wide range ofcontexts. The mathematical presentation is rigorous but its structured and comprehensivepresentation makes the information easily accessible. Clearly, it is an invaluable referenceto researchers, professionals and students dealing with functions of complex-valuedmatrices that arise frequently in many different areas. Throughout the book the examplesand exercises help the reader learn how to apply the results presented in the propositions,lemmas and theorems. In conclusion, this book provides a well organized, easy to read,authoritative and unique presentation that everyone looking to exploit complex functionsshould have available in their own shelves and libraries.

Professor Paulo S. R. Diniz, Federal University of Rio de Janeiro

Complex vector and matrix optimization problems are often encountered by researchersin the electrical engineering fields and much beyond. Their solution, which can some-times be reached from using existing standard algebra literature, may however be atime consuming and sometimes difficult process. This is particularly so when compli-cated cost function and constraint expressions arise. This book brings together severalmathematical theories in a novel manner to offer a beautifully unified and systematicmethodology for approaching such problems. It will no doubt be a great companion tomany researchers and engineers alike.

Professor David Gesbert, EURECOM, Sophia-Antipolis, France






Complex-Valued MatrixDerivativesWith Applications in Signal Processingand Communications

ARE HJØRUNGNESUniversity of Oslo, Norway






CAMBRIDGE UNIVERSITY PRESS

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Library of Congress Cataloguing in Publication dataHjørungnes, Are.Complex-Valued Matrix Derivatives : With Applications in Signal Processing and Communications /Are Hjørungnes.

p. cm.Includes bibliographical references and index.ISBN 978-0-521-19264-4 (hardback)1. Matrix derivatives. 2. Systems engineering. 3. Signal processing – Mathematical models.4. Telecommunication – Mathematical models. I. Title.TA347.D4H56 2011621.382′2 – dc22 2010046598

ISBN 978-0-521-19264-4 Hardback

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To my parents, Tove and Odd






Contents

Preface page xiAcknowledgments xiiiAbbreviations xvNomenclature xvii

1 Introduction 1

1.1 Introduction to the Book 11.2 Motivation for the Book 21.3 Brief Literature Summary 31.4 Brief Outline 5

2 Background Material 6

2.1 Introduction 62.2 Notation and Classification of Complex Variables and Functions 6

2.2.1 Complex-Valued Variables 72.2.2 Complex-Valued Functions 7

2.3 Analytic versus Non-Analytic Functions 82.4 Matrix-Related Definitions 122.5 Useful Manipulation Formulas 20

2.5.1 Moore-Penrose Inverse 232.5.2 Trace Operator 242.5.3 Kronecker and Hadamard Products 252.5.4 Complex Quadratic Forms 292.5.5 Results for Finding Generalized Matrix Derivatives 31

2.6 Exercises 38

3 Theory of Complex-Valued Matrix Derivatives 43

3.1 Introduction 433.2 Complex Differentials 44

3.2.1 Procedure for Finding Complex Differentials 463.2.2 Basic Complex Differential Properties 463.2.3 Results Used to Identify First- and Second-Order Derivatives 53






viii Contents

3.3 Derivative with Respect to Complex Matrices 553.3.1 Procedure for Finding Complex-Valued Matrix Derivatives 59

3.4 Fundamental Results on Complex-Valued Matrix Derivatives 603.4.1 Chain Rule 603.4.2 Scalar Real-Valued Functions 613.4.3 One Independent Input Matrix Variable 64

3.5 Exercises 65

4 Development of Complex-Valued Derivative Formulas 70

4.1 Introduction 704.2 Complex-Valued Derivatives of Scalar Functions 70

4.2.1 Complex-Valued Derivatives of f (z, z∗) 704.2.2 Complex-Valued Derivatives of f (z, z∗) 744.2.3 Complex-Valued Derivatives of f (Z, Z∗) 76

4.3 Complex-Valued Derivatives of Vector Functions 824.3.1 Complex-Valued Derivatives of f (z, z∗) 824.3.2 Complex-Valued Derivatives of f (z, z∗) 824.3.3 Complex-Valued Derivatives of f (Z, Z∗) 82

4.4 Complex-Valued Derivatives of Matrix Functions 844.4.1 Complex-Valued Derivatives of F(z, z∗) 844.4.2 Complex-Valued Derivatives of F(z, z∗) 854.4.3 Complex-Valued Derivatives of F(Z, Z∗) 86

4.5 Exercises 91

5 Complex Hessian Matrices for Scalar, Vector, and Matrix Functions 95

5.1 Introduction 955.2 Alternative Representations of Complex-Valued Matrix Variables 96

5.2.1 Complex-Valued Matrix Variables Z and Z∗ 965.2.2 Augmented Complex-Valued Matrix Variables Z 97

5.3 Complex Hessian Matrices of Scalar Functions 995.3.1 Complex Hessian Matrices of Scalar Functions Using Z and Z∗ 995.3.2 Complex Hessian Matrices of Scalar Functions Using Z 1055.3.3 Connections between Hessians When Using Two-Matrix

Variable Representations 1075.4 Complex Hessian Matrices of Vector Functions 1095.5 Complex Hessian Matrices of Matrix Functions 112

5.5.1 Alternative Expression of Hessian Matrix of Matrix Function 1175.5.2 Chain Rule for Complex Hessian Matrices 117

5.6 Examples of Finding Complex Hessian Matrices 1185.6.1 Examples of Finding Complex Hessian Matrices of

Scalar Functions 1185.6.2 Examples of Finding Complex Hessian Matrices of

Vector Functions 123






Contents ix

5.6.3 Examples of Finding Complex Hessian Matrices ofMatrix Functions 126

5.7 Exercises 129

6 Generalized Complex-Valued Matrix Derivatives 133

6.1 Introduction 1336.2 Derivatives of Mixture of Real- and Complex-Valued Matrix Variables 137

6.2.1 Chain Rule for Mixture of Real- and Complex-ValuedMatrix Variables 139

6.2.2 Steepest Ascent and Descent Methods for Mixture of Real- andComplex-Valued Matrix Variables 142

6.3 Definitions from the Theory of Manifolds 1446.4 Finding Generalized Complex-Valued Matrix Derivatives 147

6.4.1 Manifolds and Parameterization Function 1476.4.2 Finding the Derivative of H(X, Z, Z∗) 1526.4.3 Finding the Derivative of G(W, W∗) 1536.4.4 Specialization to Unpatterned Derivatives 1536.4.5 Specialization to Real-Valued Derivatives 1546.4.6 Specialization to Scalar Function of Square

Complex-Valued Matrices 1546.5 Examples of Generalized Complex Matrix Derivatives 157

6.5.1 Generalized Derivative with Respect to Scalar Variables 1576.5.2 Generalized Derivative with Respect to Vector Variables 1606.5.3 Generalized Matrix Derivatives with Respect to

Diagonal Matrices 1636.5.4 Generalized Matrix Derivative with Respect to

Symmetric Matrices 1666.5.5 Generalized Matrix Derivative with Respect to

Hermitian Matrices 1716.5.6 Generalized Matrix Derivative with Respect to

Skew-Symmetric Matrices 1796.5.7 Generalized Matrix Derivative with Respect to

Skew-Hermitian Matrices 1806.5.8 Orthogonal Matrices 1846.5.9 Unitary Matrices 1856.5.10 Positive Semidefinite Matrices 187

6.6 Exercises 188

7 Applications in Signal Processing and Communications 201

7.1 Introduction 2017.2 Absolute Value of Fourier Transform Example 201

7.2.1 Special Function and Matrix Definitions 2027.2.2 Objective Function Formulation 204






x Contents

7.2.3 First-Order Derivatives of the Objective Function 2047.2.4 Hessians of the Objective Function 206

7.3 Minimization of Off-Diagonal Covariance Matrix Elements 2097.4 MIMO Precoder Design for Coherent Detection 211

7.4.1 Precoded OSTBC System Model 2127.4.2 Correlated Ricean MIMO Channel Model 2137.4.3 Equivalent Single-Input Single-Output Model 2137.4.4 Exact SER Expressions for Precoded OSTBC 2147.4.5 Precoder Optimization Problem Statement and

Optimization Algorithm 2167.4.5.1 Optimal Precoder Problem Formulation 2167.4.5.2 Precoder Optimization Algorithm 217

7.5 Minimum MSE FIR MIMO Transmit and Receive Filters 2197.5.1 FIR MIMO System Model 2207.5.2 FIR MIMO Filter Expansions 2207.5.3 FIR MIMO Transmit and Receive Filter Problems 2237.5.4 FIR MIMO Receive Filter Optimization 2257.5.5 FIR MIMO Transmit Filter Optimization 226

7.6 Exercises 228

References 231Index 237






Preface

This book is written as an engineering-oriented mathematics book. It introduces the fieldinvolved in finding derivatives of complex-valued functions with respect to complex-valued matrices, in which the output of the function may be a scalar, a vector, ora matrix. The theory of complex-valued matrix derivatives, collected in this book,will benefit researchers and engineers working in fields such as signal processing andcommunications. Theories for finding complex-valued derivatives with respect to bothcomplex-valued matrices with independent components and matrices that have certaindependencies among the components are developed and illustrative examples that showhow to find such derivatives are presented. Key results are summarized in tables. Throughseveral research-related examples, it will be shown how complex-valued matrix deriva-tives can be used as a tool to solve research problems in the fields of signal processing andcommunications.

This book is suitable for M.S. and Ph.D. students, researchers, engineers, and pro-fessors working in signal processing, communications, and other fields in which theunknown variables of a problem can be expressed as complex-valued matrices. Thegoal of the book is to present the tools of complex-valued matrix derivatives suchthat the reader is able to use these theories to solve open research problems in hisor her own field. Depending on the nature of the problem, the components inside theunknown matrix might be independent, or certain interrelations might exist amongthe components. Matrices with independent components are called unpatterned and, iffunctional dependencies exist among the elements, the matrix is called patterned orstructured. Derivatives relating to complex matrices with independent components arecalled complex-valued matrix derivatives; derivatives relating to matrices that belong tosets that may contain certain structures are called generalized complex-valued matrixderivatives. Researchers and engineers can use the theories presented in this book tooptimize systems that contain complex-valued matrices. The theories in this book canbe used as tools for solving problems, with the aim of minimizing or maximizing real-valued objective functions with respect to complex-valued matrices. People who workin research and development for future signal processing and communication systemscan benefit from this book because they can use the presented material to optimize theircomplex-valued design parameters.






xii Preface

Book Overview

This book contains seven chapters. Chapter 1 gives a short introduction to thebook. Mathematical background material needed throughout the book is presented inChapter 2. Complex differentials and the definition of complex-valued derivatives areprovided in Chapter 3, and, in addition, several important theorems are proved. Chapter 4uses many examples to show the reader how complex-valued derivatives can be foundfor nine types of functions, depending on function output (scalar, vector, or matrix) andinput parameters (scalar, vector, or matrix). Second-order derivatives are presented inChapter 5, which shows how to find the Hessian matrices of complex-valued scalar,vector, and matrix functions for unpatterned matrix input variables. Chapter 6 is devotedto the theory of generalized complex-valued matrix derivatives. This theory includesderivatives with respect to complex-valued matrices that belong to certain sets, such asHermitian matrices. Chapter 7 presents several examples that show how the theory canbe used as an important tool to solve research problems related to signal processingand communications. All chapters except Chapter 1 include at least 11 exercises withrelevant problems taken from the chapters. A solution manual that provides completesolutions to problems in all exercises is available at www.cambridge.org/hjorungnes.

I will be very interested to hear from you, the reader, on any comments or suggestionsregarding this book.






Acknowledgments

During my Ph.D. studies, I started to work in the field of complex-valued matrix deriva-tives. I am very grateful to my Ph.D. advisor Professor Tor A. Ramstad at the NorwegianUniversity of Science and Technology for everything he has taught me and, in particular,for leading me onto the path to matrix derivatives. My work on matrix derivatives wasintensified when I worked as a postdoctoral Research Fellow at Helsinki University ofTechnology and the University of Oslo. The idea of writing a book developed gradually,but actual work on it started at the beginning of 2008.

I would like to thank the people at Cambridge University Press for their help. I wouldespecially like to thank Dr. Phil Meyler for the opportunity to publish this book withCambridge and Sarah Finlay, Cambridge Publishing Assistant, for her help with itspractical concerns during this preparation. Thanks also go to the reviewers of my bookproposal for helping me improve my work.

I would like to acknowledge the financial support of the Research Council of Norwayfor its funding of the FRITEK project “Theoretical Foundations of Mobile FlexibleNetworks – THEFONE” (project number 197565/V30). The THEFONE project containsone work package on complex-valued matrix derivatives.

I am grateful to Professor Zhu Han of the University of Houston for discussingwith me points on book writing and book proposals, especially during my visit to theUniversity of Houston in December, 2008. I thank Professor Paulo S. R. Diniz of theFederal University of Rio de Janeiro for helping me with questions about book proposalsand other matters relating to book writing. I am grateful to Professor David Gesbert ofEURECOM and Professor Daniel P. Palomar of Hong Kong University of Science andTechnology for their help in organizing some parts of this book and for their valuablefeedback and suggestions during its early stages. Thanks also go to Professor VisaKoivunen of the Aalto University School of Science and Technology for encouragingme to collect material on complex-valued matrix derivatives and for their valuablecomments on how to organize the material. I thank Professor Kenneth Kreutz-Delgadofor interesting discussions during my visit to the University of California, San Diego, inDecember, 2009, and pointing out several relevant references. Dr. Per Christian Moanhelped by discussing several topics in this book in an inspiring and friendly atmosphere.I am grateful to Professor Hans Brodersen and Professor John Rognes, both of theUniversity of Oslo, for discussions related to the initial material on manifold. I alsothank Professor Aleksandar Kavcic of the University of Hawai′i at Manoa for helpingarrange my sabbatical in Hawai′i from mid-July, 2010, to mid-July, 2011.






xiv Acknowledgments

Thanks go to the postdoctoral research fellows and Ph.D. students in my researchgroup, in addition to all the inspiring guests who visited with my group while I waswriting this book. Several people have helped me find errors and improve the material. Iwould especially like to thank Dr. Ninoslav Marina, who has been of great help in findingtypographical errors. I thank Professor Manav R. Bhatnagar of the Indian Institute ofTechnology Delhi; Professor Dusit Niyato of Nanyang Technological University; Dr.Xiangyun Zhou; and Dr. David K. Choi for their suggestions. In addition, thanks goto Martin Makundi, Dr. Marius Sırbu, Dr. Timo Roman, and Dr. Traian Abrudan, whomade corrections on early versions of this book.

Finally, I thank my friends and family for their support during the preparation andwriting of this book.






Abbreviations

BER bit error rateCDMA code division multiple accessCFO carrier frequency offsetDFT discrete Fourier transformFIR finite impulse responsei.i.d. independent and identically distributedLOS line-of-sightLTI linear time-invariantMIMO multiple-input multiple-outputMLD maximum likelihood decodingMSE mean square errorOFDM orthogonal frequency-division multiplexingOSTBC orthogonal space-time block codePAM pulse amplitude modulationPSK phase shift keyingQAM quadrature amplitude modulationSER symbol error rateSISO single-input single-outputSNR signal-to-noise ratioSVD singular value decompositionTDMA time division multiple accesswrt. with respect to






Nomenclature

⊗ Kronecker product� Hadamard product� defined equal to⊆ subset of⊂ proper subset of∧ logical conjunction∀ for all∑

summation∏product

× Cartesian product∫integral

≤ less than or equal to< strictly less than≥ greater than or equal to> strictly greater than� S � 0N×N means that S is positive semidefinite∞ infinity�= not equal to| such that| · | (1) |z| ≥ 0 returns the absolute value of the number z ∈ C

(2) |z| ∈ (R+ ∪ {0})N×1 returns the component-wise absolutevalues of the vector z ∈ C

N×1

(3) |A| returns the cardinality of the set A∠(·) (1) ∠z returns the principal value of the argument of the complex

input variable z(2) ∠z ∈ (−π, π ]N×1 returns the component-wise principal

argument of the vector z ∈ CN×1

∼ is statistically distributed according to0M×N M × N matrix containing only zeros1M×N M × N matrix containing only ones(·)∗ Z∗ means component-wise complex conjugation of the elements

in the matrix Z∅ empty set\ set difference






xviii Nomenclature

(·)−1 matrix inverse| · |−1 if z ∈ {C \ {0}}N×1, then |z|−1 returns a vector in (R+)N×1 with

the inverse of the component-wise absolute values of z(·)+ Moore-Penrose inverse(·)# adjoint of a matrixC set of complex numbersC(A) column space of the matrix ACN complex normally distributedN (A) null space of the matrix AR(A) row space of the matrix Aδi, j Kronecker delta function with two input argumentsδi, j,k Kronecker delta function with three input argumentsλmax(·) maximum eigenvalue of the input matrix, which must be

Hermitianλmin(·) minimum eigenvalue of the input matrix, which must be

Hermitianµ Lagrange multiplier∇Z f the gradient of f with respect to Z∗ and ∇Z f ∈ C

N×Q whenZ ∈ C

N×Q

∂

∂zformal derivative with respect to z given by ∂

∂z = 12

(∂∂x − j ∂

∂y

)∂

∂z∗ formal derivative with respect to z given by ∂∂z∗ = 1

2

(∂∂x + j ∂

∂y

)∂

∂ Zf the gradient of f with respect to Z ∈ C

N×Q and ∂∂ Z f ∈ C

N×Q

∂

∂zTf (z, z∗) formal derivatives of the vector function f : C

N×1 × CN×1

→ CM×1 with respect to the row vector zT , and

∂∂zT f (z, z∗) ∈ C

M×N

∂

∂zHf (z, z∗) formal derivatives of the vector function f : C

N×1 × CN×1

→ CM×1 with respect to the row vector zH , and

∂∂zH f (z, z∗) ∈ C

M×N

π mathematical constant, π ≈ 3.14159265358979323846ai i-th vector component of the vector aak,l (k, l)-th element of the matrix A{a0, a1, . . . , aN−1} set that contains the N elements a0, a1, . . . , aN−1

[a0, a1, . . . , aN−1] row vector of size 1 × N , where the i-th elements is given by ai

a · b, a × b a multiplied by b

‖a‖ the Euclidean norm of the vector a ∈ CN×1, i.e., ‖a‖ = √

aH aA�k the Hadamard product of A with itself k timesA−T the transposed of the inverse of the invertible square matrix A,

i.e., A−T = (A−1

)T

Ak,: k-th row of the matrix AA:,k = ak k-th column of the matrix A






Nomenclature xix

(A)k,l (k, l)-th component of the matrix A, i.e., (A)k,l = ak,l

‖A‖F the Frobenius norm of the matrix A ∈ CN×Q , i.e.,

‖A‖F =√

Tr{

AAH}

A × B Cartesian product of the two sets A and B, that is,A × B = {(a, b) | a ∈ A, b ∈ B}

arctan inverse tangentargmin minimizing argumentck,l (Z) the (k, l)-th cofactor of the matrix Z ∈ C

N×N

C(Z) if Z ∈ CN×N , then the matrix C(Z) ∈ C

N×N contains thecofactors of Z

d differential operatorDZ F complex-valued matrix derivative of the matrix function F with

respect to the matrix variable ZDN duplication matrix of size N 2 × N (N+1)

2det(·) determinant of a matrixdimC{·} complex dimension of the space it is applied todimR{·} real dimension of the space it is applied todiag(·) diagonalization operator produces a diagonal matrix from a

column vectore base of natural logarithm, e ≈ 2.71828182845904523536E[·] expected value operatorez = exp(z) complex exponential function of the complex scalar z

ej∠z if z ∈ CN×1, then ej∠z �

[ej∠z0 , ej∠z1 , . . . , ej∠zN−1

]T, where

∠zi ∈ (−π, π ] denotes the principal value of the argument of zi

exp(Z) complex exponential matrix function, which has a complex squarematrix Z as input variable

ei standard basis in CN×1

Ei, j Ei, j ∈ CN×N is given by Ei, j = ei eT

j

E Mt × (m + 1)N row-expansion of the FIR MIMOfilter {E(k)}m

k=0, where E(k) ∈ CMt ×N

E (m + 1)Mt × N column-expansion of the FIR MIMOfilter {E(k)}m


E(l) (l + 1)Mt × (m + l + 1)N matrix, which expresses therow-diagonal expanded matrix of order l of the FIR MIMOfilter {E(k)}m


E(l) (m + l + 1)Mt × (l + 1)N matrix, which expresses thecolumn-diagonal expanded matrix of order l of the FIR MIMOfilter {E(k)}m


f complex-valued scalar functionf complex-valued vector functionF complex-valued matrix functionFN N × N inverse DFT matrixf : X → Y f is a function with domain X and range Y






xx Nomenclature

(·)H AH is the conjugate transpose of the matrix AH (x) differential entropy of xH (x | y) conditional differential entropy of x when y is givenI (x; y) mutual information between x and yI identity matrixI p p × p identity matrix

I (k)N N × N matrix containing zeros everywhere and ones on the k-th

diagonal where the lower diagonal is numbered as N − 1, themain diagonal is numbered with 0, and the upper diagonal isnumbered with −(N − 1)

Im{·} returns imaginary part of the inputj imaginary unitJ M N × M N matrix with N × N identity matrices on the main

reverse block diagonal and zeros elsewhere, i.e., J = J M ⊗ I N

J N N × N reverse identity matrix with zeros everywhere except +1on the main reverse diagonal

J(k)N N × N matrix containing zeros everywhere and ones on the k-th

reverse diagonal where the upper reverse is numbered by N − 1,the main reverse diagonal is numbered with 0, and the lowerreverse diagonal is numbered with −(N − 1)

KN×Q N × Q dimensional vector space over the field K and possible

values of K are, for example, R or C

K Q,N commutation matrix of size QN × QNL Lagrange functionLd N 2 × N matrix used to place the diagonal elements of A ∈ C

N×N

on vec(A)Ll N 2 × N (N−1)

2 matrix used to place the elements strictly below themain diagonal of A ∈ C

N×N on vec(A)Lu N 2 × N (N−1)

2 matrix used to place the elements strictly above themain diagonal of A ∈ C

N×N on vec(A)limz→a

f (z) limit of f (z) when z approaches a

ln(z) principal value of natural logarithm of z, where z ∈ C

mk,l(Z) the (k, l)-th minor of the matrix Z ∈ CN×N

M(Z) if Z ∈ CN×N , then the matrix M(Z) ∈ C

N×N contains the minorsof Z

max maximum value ofmin minimum value ofN natural numbers {1, 2, 3, . . .}n! factorial of n given by n! =

n∏i=1

i = 1 · 2 · 3 · . . . · n

perm(·) permanent of a matrixP N primary circular matrix of size N × NR the set of real numbers






Nomenclature xxi

R+ the set (0,∞)

rank(·) rank of a matrixRe{·} returns real part of the input(·)T AT is the transpose of the matrix AT (k){·} linear reshaping operator used in connection with transmitter FIR

MIMO optimizationTr{·} trace of a square matrixv(·) return all the elements on and below main diagonal taken in the

same column-wise order as the ordinary vec-operatorvec(·) vectorization operator stacks the columns into a long column

vectorvecd (·) extracts the diagonal elements of a square matrix and returns them

in a column vectorvecl (·) extracts the elements strictly below the main diagonal of a square

matrix in a column-wise manner and returns them into a columnvector

vecu(·) extracts the elements strictly above the main diagonal of a squarematrix in a row-wise manner and returns them into a columnvector

vecb(·) block vectorization operator stacks square block matrices of theinput into a long block column matrix

V permutation matrix of size N (N+1)2 × N (N+1)

2 given byV = [Vd , V l]

Vd matrix of size N (N+1)2 × N used to place the elements of vecd (A)

on v(A), where A ∈ CN×N is symmetric

V l matrix of size N (N+1)2 × N (N−1)

2 used to place the elements ofvecl(A) on v(A), where A ∈ C

N×N is symmetricW set containing matrices in a manifoldW∗ set containing all the complex conjugate elements of the elements

in W , that is, when W is given, W∗ � {W∗ | W ∈ W}W symbol often used to represent a matrix in a manifold, that is,

W ∈ W , where W represents a manifoldW matrix used to represent a matrix of the same size as the matrix

W; however, the matrix W is unpatterned[x0, x1] closed interval given by the set {x | x0 ≤ x ≤ x1}(x0, x1] semi-open interval given by the set {x | x0 < x ≤ x1}(x0, x1) open interval given by the set {x | x0 < x < x1}x(n)(ν) column-expansion of vector time-series of size (ν + 1)N × 1,

where x(n) ∈ CN×1

Z the set of integers, that is, Z = {. . . ,−2,−1, 0, 1, 2, . . .}ZN the set {0, 1, . . . , N − 1}z complex-valued scalar variablez complex-valued vector variableZ complex-valued matrix variable






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