Mälardalen University Press Licentiate Theses No. 253
GENERALIZED VANDERMONDE MATRICES AND
DETERMINANTS IN ELECTROMAGNETIC COMPATIBILITY
Karl Lundengård 2017
School of Education, Culture and Communication
GENERALIZED VANDERMONDE MATRICES AND
DETERMINANTS IN ELECTROMAGNETIC COMPATIBILITY
Karl Lundengård
2017
Copyright © Karl Lundengård, 2017 ISBN 978-91-7485-312-4
ISSN 1651-9256
Printed by E-Print AB, Stockholm, Sweden
are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility. In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, appli-cations and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.
The second chapter focuses on finding the extreme points for the deter-minant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.
The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a stan-dard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic dis-charge immunity as well as some experimentally measured data of similar phenomena.
are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility. In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, appli-cations and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.
The second chapter focuses on finding the extreme points for the deter-minant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.
The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a stan-dard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic dis-charge immunity as well as some experimentally measured data of similar phenomena.
First I am very grateful to my closest colleagues. My main supervisor Pro-fessor Sergei Silvestrov introduced the Vandermonde matrix to me and sug-gested the research pursued in this thesis. From him and my co-supervisor Professor Anatoliy Malyarenko I have learned invaluable lessons about math-ematics research that will be very important in my future career. Optimising the Vandermonde determinant would have been less fruitful and engaging without fellow research student Jonas ¨Osterberg whose skills and ideas com-plement my own very well. I am very glad to have worked with Dr. Milica Ranˇci´c whose conscientiousness, work ethic and patience when introducing me to the world of electromagnetic compatibility and developing and evalu-ating the ideas used in this thesis makes me consider her a true role model for an interdisciplinary researcher. The research related to electromagnetic compatibility owes a lot to the regular input from Assistant Professor Vesna Javor to ensure the relevance and quality of the work.
I am also grateful to the other research students at M¨alardalen University that I have worked alongside with. I have found the environment at the School for Education, Culture and Communication at M¨alardalen University excellent and full of friendly, helpful and skilled co-workers.
Last but not least a very heartfelt thank you to my family for all the support, encouragement and assistance you have given me. A special men-tion to my sister for help with translating from 18th century French. I am also very sad that I will not get to spend time explaining the contents of this thesis to my father whose entire mathematics career consisted of unsuc-cessfully solving a single problem on the blackboard in 9th grade. On the other hand, my mother’s modest academic credentials have never stopped her from engaging, discussing and enjoying my work, interests and other new knowledge so I am sure she will keep me busy.
Without the ideas, requests, remarks, questions, encouragements and patience of those around me this work would not have been completed.
First I am very grateful to my closest colleagues. My main supervisor Pro-fessor Sergei Silvestrov introduced the Vandermonde matrix to me and sug-gested the research pursued in this thesis. From him and my co-supervisor Professor Anatoliy Malyarenko I have learned invaluable lessons about math-ematics research that will be very important in my future career. Optimising the Vandermonde determinant would have been less fruitful and engaging without fellow research student Jonas ¨Osterberg whose skills and ideas com-plement my own very well. I am very glad to have worked with Dr. Milica Ranˇci´c whose conscientiousness, work ethic and patience when introducing me to the world of electromagnetic compatibility and developing and evalu-ating the ideas used in this thesis makes me consider her a true role model for an interdisciplinary researcher. The research related to electromagnetic compatibility owes a lot to the regular input from Assistant Professor Vesna Javor to ensure the relevance and quality of the work.
I am also grateful to the other research students at M¨alardalen University that I have worked alongside with. I have found the environment at the School for Education, Culture and Communication at M¨alardalen University excellent and full of friendly, helpful and skilled co-workers.
Last but not least a very heartfelt thank you to my family for all the support, encouragement and assistance you have given me. A special men-tion to my sister for help with translating from 18th century French. I am also very sad that I will not get to spend time explaining the contents of this thesis to my father whose entire mathematics career consisted of unsuc-cessfully solving a single problem on the blackboard in 9th grade. On the other hand, my mother’s modest academic credentials have never stopped her from engaging, discussing and enjoying my work, interests and other new knowledge so I am sure she will keep me busy.
Without the ideas, requests, remarks, questions, encouragements and patience of those around me this work would not have been completed.
Popul¨
arvetenskaplig sammanfattning
Denna licentiatuppsats behandlar tv˚a olika ¨amnen, optimering av determi-nanten av Vandermonde-matrisen ¨over olika volymer i olika dimensioner och hur en viss klass av funktioner kan anv¨andas f¨or att approximera str¨ommen i elektrostatiska urladdningar som anv¨ands f¨or att s¨akerst¨alla elektromag-netisk kompatibilitet. En exempel p˚a kopplingen mellan de tv˚a omr˚adena ges i den sista delen av uppsatsen.
En Vandermonde-matris ¨ar en matris d¨ar raderna (eller kolonnerna) ges av stigande potenser och s˚adana matriser f¨orekommer i m˚anga olika sam-manhang, b˚ade inom abstrakt matematik och till¨ampningar inom andra omr˚aden. I uppsatsen ges en kort genomg˚ang av Vandermonde-matrisens historia och till¨ampningar av den och n˚agra besl¨aktade matriser. Fokus ligger p˚a interpolation or regression vilket kortfattat kan beskrivas som metoder f¨or att anpassa en matematisk beskrivning till given data, t.ex. fr˚an experimentella m¨atningar.
Determinanten av en matris ¨ar ett tal som ber¨aknas fr˚an matrisens el-ement och som p˚a ett kompakt s¨att kan beskriva flera olika egenskaper av matrisen eller systemet som matrisen beskriver. I denna uppsats diskuteras hur man skall v¨alja element i Vandermonde-matrisen f¨or att maximera de-terminanten under f¨oruts¨attningen att elementen i Vandermonde-matrisen tolkas som en punkt i en volym (som kan ha fler ¨an tre dimensioner). Flera volymer unders¨oks, bland annat klot, kuber, ellipsoider och torus.
En motivering till varf¨or det ¨ar anv¨andbart att veta hur Vandermonde-matrisens determinant kan maximeras ¨ar att det kan anv¨andas till optimal
experiment design, det vill s¨aga att avg¨ora hur man skall v¨alja m¨atpunkter f¨or att kunna bygga en s˚a bra matematisk modell som m¨ojligt. Ett exempel p˚a hur detta kan g˚a till ges i sista delen av uppsatsen.
Ett omr˚ade d¨ar det ¨ar anv¨andbart att kunna bygga matematiska mod-eller fr˚an experimentella data ¨ar elektromagnetisk kompatibilitet. Detta omr˚ade handlar om att s¨akerst¨alla att system som inneh˚aller elektronik inte p˚averkas f¨or mycket av externa elektromagnetiska st¨orningar eller st¨or an-dra system d˚a de anv¨ands. En viktig del av detta omr˚ade ¨ar att unders¨oka hur systemet reagerar p˚a olika externa st¨orningar s˚asom de beskrivs i olika konstruktionsstandarder. I uppsatsen diskuterar vi hur man med hj¨alp av en specifik klass av funktioner kan konstruera matematiska modeller baser-ade p˚a specifikationer i standarder eller experimentella data. V¨albekanta fenomen s˚asom blixtnedslag och urladdningar av statisk elektricitet mellan m¨anniska och metallf¨orem˚al diskuteras.
Popular-science summary
This licentiate thesis discusses two different topics, optimisation of the Van-dermonde determinant over different volumes in various dimensions and how a certain class of functions can be used to approximate the current in elec-trostatic discharges used in ensuring electromagnetic compatibility. An ex-ample of how the two topics can be connected is given in the final part of the thesis.
A Vandermonde matrix is a matrix with rows (or column) given by increasing powers and such matrices appear in many different circumstances, both in abstract mathematics and various applications. In the thesis a brief history of the Vandermonde matrix is given as well as a discussion of some applications of the Vandermonde matrix and some related matrices. The main topics will be interpolation and regression which can be described as methods for fitting a mathematical description to collected data from for example experimental measurements.
The determinant of a matrix is a number calculated from the elements of the matrix in a particular way and it can describe properties of the matrix or the system it describes in a compact way. In this thesis it is discussed how to choose the elements of the Vandermonde matrix to maximise the determinant under the constraint that the elements that define the Vander-monde determinant are interpreted as points in a certain volume (that can have a dimension higher than three). Examined volumes include spheres, cubes, ellipsoids and tori.
One way to motivate the usefulness of knowing how to maximise the Vandermonde determinant is that it can be used in optimal experiment
de-sign, that is determining how to choose the data points in an experiment to
construct the best possible mathematical model. An example of how to do this can be found in the final section of the thesis.
One area where it is useful to construct mathematical model from ex-perimental data is electromagnetic compatibility. This is the study if how to ensure that a system that contains electronics is not disturbed too much by external electromagnetic disturbances or disturbs other systems when it is used. An important aspect of this field is examining how systems respond to different external disturbances described in different construction stan-dards. In this thesis we discuss how a specific class of functions can be used to construct mathematical models based on specifications in standards or experimental data. Well-known phenomenon such as lightning strikes and electrostatic discharges from a human being to a metal object are discussed.
Popul¨
arvetenskaplig sammanfattning
Denna licentiatuppsats behandlar tv˚a olika ¨amnen, optimering av determi-nanten av Vandermonde-matrisen ¨over olika volymer i olika dimensioner och hur en viss klass av funktioner kan anv¨andas f¨or att approximera str¨ommen i elektrostatiska urladdningar som anv¨ands f¨or att s¨akerst¨alla elektromag-netisk kompatibilitet. En exempel p˚a kopplingen mellan de tv˚a omr˚adena ges i den sista delen av uppsatsen.
En Vandermonde-matris ¨ar en matris d¨ar raderna (eller kolonnerna) ges av stigande potenser och s˚adana matriser f¨orekommer i m˚anga olika sam-manhang, b˚ade inom abstrakt matematik och till¨ampningar inom andra omr˚aden. I uppsatsen ges en kort genomg˚ang av Vandermonde-matrisens historia och till¨ampningar av den och n˚agra besl¨aktade matriser. Fokus ligger p˚a interpolation or regression vilket kortfattat kan beskrivas som metoder f¨or att anpassa en matematisk beskrivning till given data, t.ex. fr˚an experimentella m¨atningar.
Determinanten av en matris ¨ar ett tal som ber¨aknas fr˚an matrisens el-ement och som p˚a ett kompakt s¨att kan beskriva flera olika egenskaper av matrisen eller systemet som matrisen beskriver. I denna uppsats diskuteras hur man skall v¨alja element i Vandermonde-matrisen f¨or att maximera de-terminanten under f¨oruts¨attningen att elementen i Vandermonde-matrisen tolkas som en punkt i en volym (som kan ha fler ¨an tre dimensioner). Flera volymer unders¨oks, bland annat klot, kuber, ellipsoider och torus.
En motivering till varf¨or det ¨ar anv¨andbart att veta hur Vandermonde-matrisens determinant kan maximeras ¨ar att det kan anv¨andas till optimal
experiment design, det vill s¨aga att avg¨ora hur man skall v¨alja m¨atpunkter f¨or att kunna bygga en s˚a bra matematisk modell som m¨ojligt. Ett exempel p˚a hur detta kan g˚a till ges i sista delen av uppsatsen.
Ett omr˚ade d¨ar det ¨ar anv¨andbart att kunna bygga matematiska mod-eller fr˚an experimentella data ¨ar elektromagnetisk kompatibilitet. Detta omr˚ade handlar om att s¨akerst¨alla att system som inneh˚aller elektronik inte p˚averkas f¨or mycket av externa elektromagnetiska st¨orningar eller st¨or an-dra system d˚a de anv¨ands. En viktig del av detta omr˚ade ¨ar att unders¨oka hur systemet reagerar p˚a olika externa st¨orningar s˚asom de beskrivs i olika konstruktionsstandarder. I uppsatsen diskuterar vi hur man med hj¨alp av en specifik klass av funktioner kan konstruera matematiska modeller baser-ade p˚a specifikationer i standarder eller experimentella data. V¨albekanta fenomen s˚asom blixtnedslag och urladdningar av statisk elektricitet mellan m¨anniska och metallf¨orem˚al diskuteras.
Popular-science summary
This licentiate thesis discusses two different topics, optimisation of the Van-dermonde determinant over different volumes in various dimensions and how a certain class of functions can be used to approximate the current in elec-trostatic discharges used in ensuring electromagnetic compatibility. An ex-ample of how the two topics can be connected is given in the final part of the thesis.
A Vandermonde matrix is a matrix with rows (or column) given by increasing powers and such matrices appear in many different circumstances, both in abstract mathematics and various applications. In the thesis a brief history of the Vandermonde matrix is given as well as a discussion of some applications of the Vandermonde matrix and some related matrices. The main topics will be interpolation and regression which can be described as methods for fitting a mathematical description to collected data from for example experimental measurements.
The determinant of a matrix is a number calculated from the elements of the matrix in a particular way and it can describe properties of the matrix or the system it describes in a compact way. In this thesis it is discussed how to choose the elements of the Vandermonde matrix to maximise the determinant under the constraint that the elements that define the Vander-monde determinant are interpreted as points in a certain volume (that can have a dimension higher than three). Examined volumes include spheres, cubes, ellipsoids and tori.
One way to motivate the usefulness of knowing how to maximise the Vandermonde determinant is that it can be used in optimal experiment
de-sign, that is determining how to choose the data points in an experiment to
construct the best possible mathematical model. An example of how to do this can be found in the final section of the thesis.
One area where it is useful to construct mathematical model from ex-perimental data is electromagnetic compatibility. This is the study if how to ensure that a system that contains electronics is not disturbed too much by external electromagnetic disturbances or disturbs other systems when it is used. An important aspect of this field is examining how systems respond to different external disturbances described in different construction stan-dards. In this thesis we discuss how a specific class of functions can be used to construct mathematical models based on specifications in standards or experimental data. Well-known phenomenon such as lightning strikes and electrostatic discharges from a human being to a metal object are discussed.
v, M - Bold, roman lower- and uppercase letters denote vectors and matrices respectively. Mi,j - Element on the ith row and jth column of M.
M·,j, Mi,· - Column (row) vector containing all elements
from the jth column (ith row) of M. [aij]nmij - n× m matrix with element aij in
the ith row and jth column. Vnm, Vn = Vnn - n× m Vandermonde matrix.
Gnm, Gn = Gnn - n× m generalized Vandermonde matrix.
Standard sets
Z, N, R, C - Sets of all integers, natural numbers (including 0), real numbers and complex numbers.
Snp, Sn = Sn2 - The n-dimensional sphere defined by the p - norm,
Snp(r) = x∈ Rn+1 n+1 k=1 |xk|p= r .
Ck[K] - All functions onK with continuous kth derivative. Special functions
Definitions can be found in standard texts.
Suggested sources use notation consistent with thesis.
Hn, Cn(α), Pn(α,β) - Hermite, Gegenbauer and Jacobi polynomials, see [2].
Γ(x), ψ(x) - The Gamma- and Digamma functions, see [2].
2F2(a, b; c; x) - The hypergeometric function, see [2].
Gm,np,q z ba
- The Meijer G-function, see [139]. Other
df
dx = f (x) - Derivative of the function f with respect to x. dkf
dxk = f(k)(x) - kth derivative of the function f with respect to x.
∂f
∂x = f (x) - Partial derivative of the function f with respect to x.
v, M - Bold, roman lower- and uppercase letters denote vectors and matrices respectively. Mi,j - Element on the ith row and jth column of M.
M·,j, Mi,· - Column (row) vector containing all elements
from the jth column (ith row) of M. [aij]nmij - n× m matrix with element aij in
the ith row and jth column. Vnm, Vn= Vnn - n× m Vandermonde matrix.
Gnm, Gn= Gnn - n× m generalized Vandermonde matrix.
Standard sets
Z, N, R, C - Sets of all integers, natural numbers (including 0), real numbers and complex numbers.
Snp, Sn= Sn2 - The n-dimensional sphere defined by the p - norm,
Snp(r) = x∈ Rn+1 n+1 k=1 |xk|p= r .
Ck[K] - All functions onK with continuous kth derivative. Special functions
Definitions can be found in standard texts.
Suggested sources use notation consistent with thesis.
Hn, Cn(α), Pn(α,β) - Hermite, Gegenbauer and Jacobi polynomials, see [2].
Γ(x), ψ(x) - The Gamma- and Digamma functions, see [2].
2F2(a, b; c; x) - The hypergeometric function, see [2].
Gm,np,q z ab
- The Meijer G-function, see [139]. Other
df
dx = f (x) - Derivative of the function f with respect to x. dkf
dxk = f(k)(x) - kth derivative of the function f with respect to x.
∂f
∂x = f (x) - Partial derivative of the function f with respect to x.
List of Papers 13
1 Introduction 15
1.1 The Vandermonde matrix . . . 19
1.1.1 Who was Vandermonde? . . . 19
1.1.2 The Vandermonde determinant . . . 21
1.1.3 Inverse of the Vandermonde matrix . . . 26
1.1.4 The alternant matrix . . . 27
1.1.5 The generalized Vandermonde matrix . . . 30
1.2 Interpolation . . . 32
1.2.1 Polynomial interpolation . . . 33
1.3 Regression . . . 37
1.3.1 Linear regression models . . . 38
1.3.2 Non-linear regression models . . . 39
1.3.3 The Marquardt least-squares method . . . 39
1.3.4 D-optimal experiment design . . . . 43
1.4 Electromagnetic compatibility and electrostatic discharge currents . . . 46
1.4.1 Electrostatic discharge modelling . . . 48
List of Papers 13
1 Introduction 15
1.1 The Vandermonde matrix . . . 19
1.1.1 Who was Vandermonde? . . . 19
1.1.2 The Vandermonde determinant . . . 21
1.1.3 Inverse of the Vandermonde matrix . . . 26
1.1.4 The alternant matrix . . . 27
1.1.5 The generalized Vandermonde matrix . . . 30
1.2 Interpolation . . . 32
1.2.1 Polynomial interpolation . . . 33
1.3 Regression . . . 37
1.3.1 Linear regression models . . . 38
1.3.2 Non-linear regression models . . . 39
1.3.3 The Marquardt least-squares method . . . 39
1.3.4 D-optimal experiment design . . . . 43
1.4 Electromagnetic compatibility and electrostatic discharge currents . . . 46
1.4.1 Electrostatic discharge modelling . . . 48
2 Extreme points of the Vandermonde determinant 53 2.1 Extreme points of the Vandermonde determinant and related
determinants on various surfaces in three dimensions . . . 55
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . 55
2.1.2 Extreme points of the Vandermonde determinant on the three-dimensional unit sphere . . . 59
2.1.3 Optimisation of the Vandermonde determinant on the three-dimensional torus . . . 60
2.1.4 Optimisation using Gr¨obner bases . . . 64
2.1.5 Extreme points on the ellipsoid in three dimensions . 66 2.1.6 Extreme points on the cylinder in three dimensions . . 68
2.1.7 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . 70
2.2 Optimization of the Vandermonde determinant on some n-dimensional surfaces . . . . 72
2.2.1 The extreme points on the sphere given by roots of a polynomial . . . 73
2.2.2 Further visual exploration on the sphere . . . 81
2.2.3 The extreme points on spheres defined by the p-norms given by roots of a polynomial . . . 88
2.2.4 The Vandermonde determinant on spheres defined by the 4-norm . . . 96
3 Approximation of electrostatic discharge currents using the analytically extended function 99 3.1 The analytically extended function (AEF) . . . 101
3.1.1 The p-peak analytically extended function . . . 102
3.2 Approximation of lightning discharge current functions . . . . 109
3.2.1 Fitting the AEF . . . 110
3.2.2 Estimating parameters for underdetermined systems . 111 3.2.3 Fitting with data points as well as charge flow and specific energy conditions . . . 112
3.2.4 Calculating the η-parameters from the β-parameters . 115 3.2.5 Explicit formulas for a single-peak AEF . . . 116
3.2.6 Fitting to lightning discharge currents . . . 117
3.3 Approximation of electrostatic discharge currents . . . 121
3.3.1 IEC 61000-4-2 Standard current waveshape . . . 122
3.3.2 D-Optimal approximation for exponents given by a class of arithmetic sequences . . . 125
3.3.3 Examples of models from applications and experiments 129 3.3.4 Summary of ESD modelling . . . 131
References 135 Index 151 List of Figures 153 List of Tables 156 List of Definitions 156 List of Theorems 157 List of Lemmas 157
2 Extreme points of the Vandermonde determinant 53 2.1 Extreme points of the Vandermonde determinant and related
determinants on various surfaces in three dimensions . . . 55
2.1.1 Optimization of the generalized Vandermonde deter-minant in three dimensions . . . 55
2.1.2 Extreme points of the Vandermonde determinant on the three-dimensional unit sphere . . . 59
2.1.3 Optimisation of the Vandermonde determinant on the three-dimensional torus . . . 60
2.1.4 Optimisation using Gr¨obner bases . . . 64
2.1.5 Extreme points on the ellipsoid in three dimensions . 66 2.1.6 Extreme points on the cylinder in three dimensions . . 68
2.1.7 Optimizing the Vandermonde determinant on a sur-face defined by a homogeneous polynomial . . . 70
2.2 Optimization of the Vandermonde determinant on some n-dimensional surfaces . . . . 72
2.2.1 The extreme points on the sphere given by roots of a polynomial . . . 73
2.2.2 Further visual exploration on the sphere . . . 81
2.2.3 The extreme points on spheres defined by the p-norms given by roots of a polynomial . . . 88
2.2.4 The Vandermonde determinant on spheres defined by the 4-norm . . . 96
3 Approximation of electrostatic discharge currents using the analytically extended function 99 3.1 The analytically extended function (AEF) . . . 101
3.1.1 The p-peak analytically extended function . . . 102
3.2 Approximation of lightning discharge current functions . . . . 109
3.2.1 Fitting the AEF . . . 110
3.2.2 Estimating parameters for underdetermined systems . 111 3.2.3 Fitting with data points as well as charge flow and specific energy conditions . . . 112
3.2.4 Calculating the η-parameters from the β-parameters . 115 3.2.5 Explicit formulas for a single-peak AEF . . . 116
3.2.6 Fitting to lightning discharge currents . . . 117
3.3 Approximation of electrostatic discharge currents . . . 121
3.3.1 IEC 61000-4-2 Standard current waveshape . . . 122
3.3.2 D-Optimal approximation for exponents given by a class of arithmetic sequences . . . 125
3.3.3 Examples of models from applications and experiments 129 3.3.4 Summary of ESD modelling . . . 131
References 135 Index 151 List of Figures 153 List of Tables 156 List of Definitions 156 List of Theorems 157 List of Lemmas 157
Paper A. Karl Lundeng˚ard, Jonas ¨Osterberg and Sergei Silvestrov. Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant.
Preprint: arXiv:1312.6193 [math.ca], 2013.
Paper B. Karl Lundeng˚ard, Jonas ¨Osterberg, and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix and related matrices. In
AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, pages 627–
636, 2014.
Paper C. Karl Lundeng˚ard, Jonas ¨Osterberg, and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix on the sphere and related sur-faces. In Christos H Skiadas, editor, ASMDA 2015 Proceedings: 16th Applied
Stochastic Models and Data Analysis International Conference with 4th De-mographics 2015 Workshop, pages 637–648. ISAST: International Society for
the Advancement of Science and Technology, 2015.
Paper D. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approxima-tion of lightning discharge currents. Sergei Silvestrov and Milica Ranˇci´c, editors, Engineering Mathematics I: Electromagnetics, Fluid Mechanics,
Ma-terial Physics and Financial Engineering, volume 178 of Springer Proceedings in Mathematics & Statistics. Springer International Publishing, 2016.
Paper E. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Estima-tion of parameters for the multi-peaked AEF current funcEstima-tions. Methodology
and Computing in Applied Probability, pages 1–15, 2016.
Paper F. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Electro-static discharge currents representation using the multi-peaked analytically extended function by interpolation on a D-optimal design.
Paper A. Karl Lundeng˚ard, Jonas ¨Osterberg and Sergei Silvestrov. Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant.
Preprint: arXiv:1312.6193 [math.ca], 2013.
Paper B. Karl Lundeng˚ard, Jonas ¨Osterberg, and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix and related matrices. In
AIP Conference Proceedings 1637, ICNPAA, Narvik, Norway, pages 627–
636, 2014.
Paper C. Karl Lundeng˚ard, Jonas ¨Osterberg, and Sergei Silvestrov. Optimization of the determinant of the Vandermonde matrix on the sphere and related sur-faces. In Christos H Skiadas, editor, ASMDA 2015 Proceedings: 16th Applied
Stochastic Models and Data Analysis International Conference with 4th De-mographics 2015 Workshop, pages 637–648. ISAST: International Society for
the Advancement of Science and Technology, 2015.
Paper D. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approxima-tion of lightning discharge currents. Sergei Silvestrov and Milica Ranˇci´c, editors, Engineering Mathematics I: Electromagnetics, Fluid Mechanics,
Ma-terial Physics and Financial Engineering, volume 178 of Springer Proceedings in Mathematics & Statistics. Springer International Publishing, 2016.
Paper E. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Estima-tion of parameters for the multi-peaked AEF current funcEstima-tions. Methodology
and Computing in Applied Probability, pages 1–15, 2016.
Paper F. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. Electro-static discharge currents representation using the multi-peaked analytically extended function by interpolation on a D-optimal design.
Parts of the thesis have been presented at the following international conferences: 1. 10th International Conference on Mathematical Problems in Engineering,
Aerospace and Sciences - ICNPAA 2014, Narvik, Norway, July 15-18, 2014. 2. 16th Applied Stochastic Models and Data Analysis International Conference
with 4th Demographics 2015 Workshop - ASMDA 2015, Piraeus, Greece, June 30 - July 4, 2015.
Summaries of papers A-F with a brief description of the thesis authors contributions to each paper can be found in Section 1.5.
Chapter 1
Introduction
This chapter is partially based on Paper D:
Paper D. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Sergei Silvestrov and Milica Ranˇci´c, editors, Engineering Mathematics I:
Electromag-netics, Fluid Mechanics, Material Physics and Financial Engineer-ing, volume 178 of Springer Proceedings in Mathematics & Statistics.
Parts of the thesis have been presented at the following international conferences: 1. 10th International Conference on Mathematical Problems in Engineering,
Aerospace and Sciences - ICNPAA 2014, Narvik, Norway, July 15-18, 2014. 2. 16th Applied Stochastic Models and Data Analysis International Conference
with 4th Demographics 2015 Workshop - ASMDA 2015, Piraeus, Greece, June 30 - July 4, 2015.
Summaries of papers A-F with a brief description of the thesis authors contributions to each paper can be found in Section 1.5.
Chapter 1
Introduction
This chapter is partially based on Paper D:
Paper D. Karl Lundeng˚ard, Milica Ranˇci´c, Vesna Javor and Sergei Silvestrov. On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents. Sergei Silvestrov and Milica Ranˇci´c, editors, Engineering Mathematics I:
Electromag-netics, Fluid Mechanics, Material Physics and Financial Engineer-ing, volume 178 of Springer Proceedings in Mathematics & Statistics.
The Vandermonde matrix is a well-known type of matrix that appears in many different areas. In this thesis we will discuss this matrix and some of its properties, specifically the extreme points of the determinant on various surfaces and we will use a generalized Vandermonde matrix for fitting a certain type of curve to data taken from sources that are important when analysing electromagnetic compatibility.
This thesis is based on the six papers listed on page 13. The contents have been rearranged to clarify the relations between the material in the different papers. If a section is based on a paper this is specified at the beginning of the section and unless otherwise specified any subsection is from the same source. A section that is based on a paper consists of text from the paper unchanged except for modifications to correct misprints and preserve consistency within the thesis. Parts of several papers have also been omitted to avoid repetition and improve cohesion. The relations between contents of the the sections are of many kinds, common definitions and dependent results, conceptual connections as well as similarities in proof techniques and problem formulations. This is illustrated in Figure 1.1. A reader only interested in a particular section or in a hurry can consult Figure 1.2 to find a short route to the desired content.
In chapter 1 the basic theory for later chapters is introduced. The Van-dermonde matrix and some of its properties, history, applications and gen-eralizations are briefly introduced in Section 1.1. In Section 1.2 interpola-tion problems and their relainterpola-tions to alternant- and Vandermonde matrices is described. In Section 1.3 various regression models and the Marquardt least-squares method for non-linear regression problems are discussed. The optimal design of experiments with respect to regression is discussed as well. Section 1.4 introduces electromagnetic compatibility and certain curve-fitting problems that appear in modelling of electrostatic discharge currents. Chapter 2 discusses the optimisation of the Vandermonde determinant over various surfaces. First the extreme points on the sphere in three dimen-sions are examined, Section 2.1.1 and 2.1.2. Further discussion includes the torus, cylinder and ellipsoid, Section 2.1.3–2.1.7. In Section 2.2 the determi-nant is optimised on the sphere and related surfaces in higher dimensions.
Chapter 3 discusses fitting a piecewise non-linear regression model to data. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least-squares method is de-scribed in Section 3.2.1–3.2.5. The framework is then applied to lightning discharge currents in Section 3.2.6. An alternate curve-fitting method based on D-optimal interpolation (found analogously to the results in Section 2.2) is described and applied to electrostatic discharge currents in Section 3.3.
The Vandermonde matrix is a well-known type of matrix that appears in many different areas. In this thesis we will discuss this matrix and some of its properties, specifically the extreme points of the determinant on various surfaces and we will use a generalized Vandermonde matrix for fitting a certain type of curve to data taken from sources that are important when analysing electromagnetic compatibility.
This thesis is based on the six papers listed on page 13. The contents have been rearranged to clarify the relations between the material in the different papers. If a section is based on a paper this is specified at the beginning of the section and unless otherwise specified any subsection is from the same source. A section that is based on a paper consists of text from the paper unchanged except for modifications to correct misprints and preserve consistency within the thesis. Parts of several papers have also been omitted to avoid repetition and improve cohesion. The relations between contents of the the sections are of many kinds, common definitions and dependent results, conceptual connections as well as similarities in proof techniques and problem formulations. This is illustrated in Figure 1.1. A reader only interested in a particular section or in a hurry can consult Figure 1.2 to find a short route to the desired content.
In chapter 1 the basic theory for later chapters is introduced. The Van-dermonde matrix and some of its properties, history, applications and gen-eralizations are briefly introduced in Section 1.1. In Section 1.2 interpola-tion problems and their relainterpola-tions to alternant- and Vandermonde matrices is described. In Section 1.3 various regression models and the Marquardt least-squares method for non-linear regression problems are discussed. The optimal design of experiments with respect to regression is discussed as well. Section 1.4 introduces electromagnetic compatibility and certain curve-fitting problems that appear in modelling of electrostatic discharge currents. Chapter 2 discusses the optimisation of the Vandermonde determinant over various surfaces. First the extreme points on the sphere in three dimen-sions are examined, Section 2.1.1 and 2.1.2. Further discussion includes the torus, cylinder and ellipsoid, Section 2.1.3–2.1.7. In Section 2.2 the determi-nant is optimised on the sphere and related surfaces in higher dimensions.
Chapter 3 discusses fitting a piecewise non-linear regression model to data. The particular model is introduced in Section 3.1 and a general frame-work for fitting it to data using the Marquardt least-squares method is de-scribed in Section 3.2.1–3.2.5. The framework is then applied to lightning discharge currents in Section 3.2.6. An alternate curve-fitting method based on D-optimal interpolation (found analogously to the results in Section 2.2) is described and applied to electrostatic discharge currents in Section 3.3.
Section 1.1 Section 1.1.1 Section 1.1.2 Section 1.1.3 Section 1.1.4 Section 1.1.5 Section 1.2 Section 1.2.1 Section 1.3 Section 1.3.1 Section 1.3.2 Section 1.3.3
Paper D Section 1.4 Section 1.3.4
Section 2.1.1 Paper A Sections 2.1.2-2.1.7 Paper A–C Section 2.2 Paper A–C Section 3.1 Paper D Section 3.2 Paper E Section 3.3 Paper F
Figure 1.1: Relations between sections of the thesis. Arrows indicate that the target section uses some definition or theorem from the source section. Dashed lines indicates a tangential or conceptual relation.
Section 1.1 Section 1.3 Section 1.1.2 Sections 1.1.4, 1.2.1 Section 1.1.5 Section 1.3.2 Sections 1.3.1, 1.3.4 Section 1.3.3 Paper D Section 2.1.1 Paper A Sections 2.1.2-2.1.7, 2.2 Paper A–C Section 3.3 Paper F Sections 1.4, 3.1 Paper D Section 3.2 Paper E
Figure 1.2: Reference that demonstrates short routes to the different chapters.
1.1
The Vandermonde matrix
The Vandermonde matrix is a well-known matrix with a very special form that appears in many different circumstances, a few examples are poly-nomial interpolation (see Section 1.2.1), least square regression (see Sec-tion 1.3), optimal experiment design (see SecSec-tion 1.3.4), construcSec-tion of error-detecting and error-correcting codes (see [18, 69, 143] as well as more recent work such as [17]), determining if a market with a finite set of traded assets is complete [32], calculation of the discrete Fourier trans-form [142] and related transtrans-forms such as the fractional discrete Fourier transform [122], the quantum Fourier transform [36], and the Vandermonde transform [6, 7], solving systems of differential equations with constant co-efficients [120], various problems in mathematical- [168], nuclear- [29], and quantum physics [148, 159] and describing properties of the Fisher informa-tion matrix of stainforma-tionary stochastic processes [93].
In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.
Definition 1.1. A Vandermonde matrix is an n× m matrix of the form
Vmn(xn) = xij−1 m,n i,j = 1 1 · · · 1 x1 x2 · · · xn .. . ... . .. ... xm1−1 xm2−1 · · · xmn−1 (1)
where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notation
Vn= Vnm will be used.
Remark 1.1. Note that in the literature the term Vandermonde matrix is often used for the transpose of the matrix given in expression (1).
1.1.1
Who was Vandermonde?
The matrix is named after Alexandre Th´eophile Vandermonde (1735–1796) who had a varied career that began with law studies and some success as a concert violinist, transitioned into work in science and mathematics in the beginning of the 1770s that gradually turned into administrative and leadership positions at various Parisian institutions as well as work in politics and economics in the end of the 1780s [42]. His entire mathematical career consisted of four published papers, first presented to the French Academy of Sciences in 1770 and 1771 and published a few years later.
Section 1.1 Section 1.1.1 Section 1.1.2 Section 1.1.3 Section 1.1.4 Section 1.1.5 Section 1.2 Section 1.2.1 Section 1.3 Section 1.3.1 Section 1.3.2 Section 1.3.3
Paper D Section 1.4 Section 1.3.4
Section 2.1.1 Paper A Sections 2.1.2-2.1.7 Paper A–C Section 2.2 Paper A–C Section 3.1 Paper D Section 3.2 Paper E Section 3.3 Paper F
Figure 1.1: Relations between sections of the thesis. Arrows indicate that the target section uses some definition or theorem from the source section. Dashed lines indicates a tangential or conceptual relation.
Section 1.1 Section 1.3 Section 1.1.2 Sections 1.1.4, 1.2.1 Section 1.1.5 Section 1.3.2 Sections 1.3.1, 1.3.4 Section 1.3.3 Paper D Section 2.1.1 Paper A Sections 2.1.2-2.1.7, 2.2 Paper A–C Section 3.3 Paper F Sections 1.4, 3.1 Paper D Section 3.2 Paper E
Figure 1.2: Reference that demonstrates short routes to the different chapters.
1.1
The Vandermonde matrix
The Vandermonde matrix is a well-known matrix with a very special form that appears in many different circumstances, a few examples are poly-nomial interpolation (see Section 1.2.1), least square regression (see Sec-tion 1.3), optimal experiment design (see SecSec-tion 1.3.4), construcSec-tion of error-detecting and error-correcting codes (see [18, 69, 143] as well as more recent work such as [17]), determining if a market with a finite set of traded assets is complete [32], calculation of the discrete Fourier trans-form [142] and related transtrans-forms such as the fractional discrete Fourier transform [122], the quantum Fourier transform [36], and the Vandermonde transform [6, 7], solving systems of differential equations with constant co-efficients [120], various problems in mathematical- [168], nuclear- [29], and quantum physics [148, 159] and describing properties of the Fisher informa-tion matrix of stainforma-tionary stochastic processes [93].
In this section we will review some of the basic properties of the Van-dermonde matrix, starting with its definition.
Definition 1.1. A Vandermonde matrix is an n× m matrix of the form
Vmn(xn) = xij−1 m,n i,j = 1 1 · · · 1 x1 x2 · · · xn .. . ... . .. ... xm1−1 xm2−1 · · · xmn−1 (1)
where xi ∈ C, i = 1, . . . , n. If the matrix is square, n = m, the notation
Vn= Vnm will be used.
Remark 1.1. Note that in the literature the term Vandermonde matrix is often used for the transpose of the matrix given in expression (1).
1.1.1
Who was Vandermonde?
The matrix is named after Alexandre Th´eophile Vandermonde (1735–1796) who had a varied career that began with law studies and some success as a concert violinist, transitioned into work in science and mathematics in the beginning of the 1770s that gradually turned into administrative and leadership positions at various Parisian institutions as well as work in politics and economics in the end of the 1780s [42]. His entire mathematical career consisted of four published papers, first presented to the French Academy of Sciences in 1770 and 1771 and published a few years later.
The first paper, M´emoire sur la r´esolution des ´equations [164], discusses some properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the pow-ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [99]), but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [97]. He also notices the equality
a2b + b2c + ac2− a2c− ab2− bc2 = (a− b)(a − c)(b − c), which is a special case
of the formula for the determinant of the Vandermonde matrix. It seems that Vandermonde did not understand the significance of the expression.
The second paper, Remarques sur des probl`emes de situation [165], dis-cusses the problem of the knight’s tour (what sequence of moves allows a knight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is now called knot theory [140].
The third paper, M´emoire sur des irrationnelles de diff´erents ordres avec
une application au cercle [166], is a paper on combinatorics and the most
well-known result from the paper is the Chu-Vandermonde identity,
n k=1 k j=1 r + 1− j j n −k j=1 s + 1− j j = n j=1 r + s + 1− j j ,
where r, s∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh
ca 1260 – ca 1320, traditional chinese: 朱世傑 in 1303 in The precious
mirror of the four elements 四元玉 and was rediscovered (apparently independently) by Vandermonde [4, 127].
In the fourth paper M´emoire sur l’´elimination [167] Vandermonde dis-cusses some ideas for what we today call determinants, which is functions that can tell us if a linear equation system has a unique solution or not. The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems using alternating functions, which has strong relation to determinants. He also notices that exchanging exponents for indices in a class of expressions from his first paper will give a class of expressions that he discusses in his fourth paper [179]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the deterdeter-minant of a general matrix described in Theorem 1.3.
While Vandermonde’s papers can be said to contain many important ideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician, especiusu-ally compared
to well-known mathematicians such as ´Etienne B´ezout (1730 – 1783) and Pierre-Simon de Laplace (1749 – 1827) as well as the chemist Antoine Lavoisier (1743 – 1794) that he worked with for some time after his math-ematical career. The Vandermonde matrix does not appear in any of Van-dermonde’s published works, which is not surprising considering that the modern matrix concept did not really take shape until almost a hundred years later in the works of Sylvester and Cayley [25, 157]. It is therefore strange that the Vandermonde matrix was named after him, a thorough discussion on this can be found in [179], but a possible reason is the simple formula for the determinant that Vandermonde briefly discusses in his fourth paper can be generalized to a Vandermonde matrix of any size. One of the main reasons that the Vandermonde matrix has become known is that it has an exceptionally simple expression for its determinant that in turn has a surprisingly fundamental relation to the determinant of a general matrix. We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section will introduce it and some of its properties.
1.1.2
The Vandermonde determinant
Often it is not the Vandermonde matrix itself that is useful, instead it is the multivariate polynomial given by its determinant that is examined and used. The determinant of the Vandermonde matrix is usually called the
Vander-monde determinant (or VanderVander-monde polynomial or Vandermondian [168])
and can be written using an exceptionally simple formula. But before we discuss the Vandermonde determinant we will disuss the general determi-nant.
Definition 1.2. The determinant is a function of square matrices over a field F to the field F, det : Mn×n(F) → F such that if we consider the
determinant as a function of the columns
det(M) = det(M·,1, M·,2, . . . , M·,n)
of the matrix the determinant must have the following properties
• The determinant must be multilinear
det(M·,1, . . . , aM·,k+ bN·,k, . . . , M·,n)
= a det(M·,1, . . . , M·,k, . . . , M·,n) + b det(M·,1, . . . , N·,k, . . . , M·,n).
• The determinant must be alternating, that is if M·,i= M·,j for some
The first paper, M´emoire sur la r´esolution des ´equations [164], discusses some properties of the roots of polynomial equations, more specifically for-mulas for the sum of the roots and a sum of symmetric functions of the pow-ers of the roots. This paper has been mentioned as important since it con-tains some of the fundamental ideas of group theory (see for instance [99]), but generally this work is overshadowed by the works of the contempo-rary Joseph Louis Lagrange (1736–1813) [97]. He also notices the equality
a2b + b2c + ac2− a2c− ab2− bc2= (a− b)(a − c)(b − c), which is a special case
of the formula for the determinant of the Vandermonde matrix. It seems that Vandermonde did not understand the significance of the expression.
The second paper, Remarques sur des probl`emes de situation [165], dis-cusses the problem of the knight’s tour (what sequence of moves allows a knight to visit all squares on a chessboard exactly once). This paper is con-sidered the first mathematical paper that uses the basic ideas of what is now called knot theory [140].
The third paper, M´emoire sur des irrationnelles de diff´erents ordres avec
une application au cercle [166], is a paper on combinatorics and the most
well-known result from the paper is the Chu-Vandermonde identity,
n k=1 k j=1 r + 1− j j n −k j=1 s + 1− j j = n j=1 r + s + 1− j j ,
where r, s∈ R and n ∈ Z. The identity was first found by Chu Shih-Chieh
ca 1260 – ca 1320, traditional chinese: 朱世傑 in 1303 in The precious
mirror of the four elements 四元玉 and was rediscovered (apparently independently) by Vandermonde [4, 127].
In the fourth paper M´emoire sur l’´elimination [167] Vandermonde dis-cusses some ideas for what we today call determinants, which is functions that can tell us if a linear equation system has a unique solution or not. The paper predates the modern definitions of determinants but Vander-monde discusses a general method for solving linear equation systems using alternating functions, which has strong relation to determinants. He also notices that exchanging exponents for indices in a class of expressions from his first paper will give a class of expressions that he discusses in his fourth paper [179]. This relation is mirrored in the relationship between the deter-minant of the Vandermonde matrix and the deterdeter-minant of a general matrix described in Theorem 1.3.
While Vandermonde’s papers can be said to contain many important ideas they do not bring any of them to maturity and he is therefore usu-ally considered a minor scientist and mathematician, especiusu-ally compared
to well-known mathematicians such as ´Etienne B´ezout (1730 – 1783) and Pierre-Simon de Laplace (1749 – 1827) as well as the chemist Antoine Lavoisier (1743 – 1794) that he worked with for some time after his math-ematical career. The Vandermonde matrix does not appear in any of Van-dermonde’s published works, which is not surprising considering that the modern matrix concept did not really take shape until almost a hundred years later in the works of Sylvester and Cayley [25, 157]. It is therefore strange that the Vandermonde matrix was named after him, a thorough discussion on this can be found in [179], but a possible reason is the simple formula for the determinant that Vandermonde briefly discusses in his fourth paper can be generalized to a Vandermonde matrix of any size. One of the main reasons that the Vandermonde matrix has become known is that it has an exceptionally simple expression for its determinant that in turn has a surprisingly fundamental relation to the determinant of a general matrix. We will be taking a closer look at the determinant of the Vandermonde ma-trix and related matrices several times in this thesis so the next section will introduce it and some of its properties.
1.1.2
The Vandermonde determinant
Often it is not the Vandermonde matrix itself that is useful, instead it is the multivariate polynomial given by its determinant that is examined and used. The determinant of the Vandermonde matrix is usually called the
Vander-monde determinant (or VanderVander-monde polynomial or Vandermondian [168])
and can be written using an exceptionally simple formula. But before we discuss the Vandermonde determinant we will disuss the general determi-nant.
Definition 1.2. The determinant is a function of square matrices over a field F to the field F, det : Mn×n(F) → F such that if we consider the
determinant as a function of the columns
det(M) = det(M·,1, M·,2, . . . , M·,n)
of the matrix the determinant must have the following properties
• The determinant must be multilinear
det(M·,1, . . . , aM·,k+ bN·,k, . . . , M·,n)
= a det(M·,1, . . . , M·,k, . . . , M·,n) + b det(M·,1, . . . , N·,k, . . . , M·,n).
• The determinant must be alternating, that is if M·,i= M·,j for some
• If I is the identity matrix then det(I) = 1.
Remark 1.2. Defining the multilinear and alternating properties from the rows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearity implies that switching places between two columns changes the sign of the determinant.
This definition of the determinant is quite abstract but it is sufficient to define a unique function.
Theorem 1.1 (Leibniz formula for determinants). A standard result from
linear algebra says that the determinant is unique and that it is given by the following formula det(M) = σ∈Sn (−1)I(σ) n i=1 mi,σ(i) (2)
where Sn is the set of all permutations of the set{1, 2, . . . , n}, that is all lists
that contain the numbers 1, 2, . . . , n exactly once, and if σ is a permutation then σ(i) is the ith element of that permutation.
Remark 1.3. Often formula (2) is used immediately as the definition of the determinant of a matrix, see for instance [5]. The formula is usually attributed to Gottfried Wilhem Leibniz (1646–1716), probably due to a letter that he wrote to Guillaume de l’Hˆopital (1661–1704) in 1693 where he describes a method of solving linear equation systems that is closely related to Cramer’s rule [123], the particular letter was published in [100] and a translation can be found in [153].
The determinant has several uses and interpretation, the two most well-known ones are
• If det(M) = 0 then the vectors corresponding to the columns (or
rows) are linearly independent. Compare this to the properties of the Wronskian matrix described on page 28.
• If the columns (or rows) of M are interpreted as sides defining an n-dimensional parallelepiped the absolute value of det(M) will give the
volume of this parallelepiped. Compare this to the interpretation of
D-optimality on page 43. The sign of the determinant is also important
when considering the orientation of the surface which is highly relevant in geometric algebra and integration over several variables, see for instance [112, 146] for examples.
We will now discuss the Vandermonde determinant specifically.
Theorem 1.2. The Vandermonde determinant, vn(x1, . . . , xn), is given by
vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) =
1≤i<j≤n
(xj− xi).
A simple way to phrase Theorem 1.2 is that the Vandermonde determi-nant is the product of all differences of the values that define the elements (note that this does not give the sign of the determinant).
There are several proofs of Theorem 1.2. Many are based on a com-bination of using elementary row or column operations together with in-duction [95, 117] but there are also several proofs using combinatorial [11] or graph-based techniques [60, 141]. Here we will present a simple proof, that dates back to some very early results on determinants [24] and has an interesting connection to the general concept of a determinant.
Proof of Theorem 1.2. There are many versions of this proof, see for
exam-ple [10, 21, 24, 71], with focus on different aspects of the proof. Here we will provide a fairly concise version that still makes all the steps of the proof clear. We start by only considering one of the variables xk, which gives a
single variable function vn(xk). From the general expression for the
deter-minant (Theorem 1.1) it is clear that vn(xk) must be a polynomial of degree
n in xk. We also know that if we let xk = xi for any 1≤ i ≤ n, i = k, the
determinant will be equal to zero since the corresponding matrix will have two identical columns. Thus if vn(xi) = 0 we can write
vn(xk) = P (xk) n i=1 i =k (xk− xi)
where P (xk) is a polynomial. If we repeat this argument for all the variables,
and ensure that no roots appear twice in the factorization, we get
vn(x1, . . . , xn) = Pn(x1, . . . , xn) n −1 i=1 (xn− xi) = Pn−1(x1, . . . , xn) n −2 i=1 (xn−1− xi) n −1 i=1 (xn− xi) = P0(x1, . . . , xn)(x2− x1)(x3− x2)(x3− x1)· · · n −1 i=1 (xn− xi)
• If I is the identity matrix then det(I) = 1.
Remark 1.2. Defining the multilinear and alternating properties from the rows of the matrix will give the same determinant. The name of the alter-nating property comes from the fact that it combined with multilinearity implies that switching places between two columns changes the sign of the determinant.
This definition of the determinant is quite abstract but it is sufficient to define a unique function.
Theorem 1.1 (Leibniz formula for determinants). A standard result from
linear algebra says that the determinant is unique and that it is given by the following formula det(M) = σ∈Sn (−1)I(σ) n i=1 mi,σ(i) (2)
where Snis the set of all permutations of the set{1, 2, . . . , n}, that is all lists
that contain the numbers 1, 2, . . . , n exactly once, and if σ is a permutation then σ(i) is the ith element of that permutation.
Remark 1.3. Often formula (2) is used immediately as the definition of the determinant of a matrix, see for instance [5]. The formula is usually attributed to Gottfried Wilhem Leibniz (1646–1716), probably due to a letter that he wrote to Guillaume de l’Hˆopital (1661–1704) in 1693 where he describes a method of solving linear equation systems that is closely related to Cramer’s rule [123], the particular letter was published in [100] and a translation can be found in [153].
The determinant has several uses and interpretation, the two most well-known ones are
• If det(M) = 0 then the vectors corresponding to the columns (or
rows) are linearly independent. Compare this to the properties of the Wronskian matrix described on page 28.
• If the columns (or rows) of M are interpreted as sides defining an n-dimensional parallelepiped the absolute value of det(M) will give the
volume of this parallelepiped. Compare this to the interpretation of
D-optimality on page 43. The sign of the determinant is also important
when considering the orientation of the surface which is highly relevant in geometric algebra and integration over several variables, see for instance [112, 146] for examples.
We will now discuss the Vandermonde determinant specifically.
Theorem 1.2. The Vandermonde determinant, vn(x1, . . . , xn), is given by
vn(x1, . . . , xn) = det(Vn(x1, . . . , xn)) =
1≤i<j≤n
(xj− xi).
A simple way to phrase Theorem 1.2 is that the Vandermonde determi-nant is the product of all differences of the values that define the elements (note that this does not give the sign of the determinant).
There are several proofs of Theorem 1.2. Many are based on a com-bination of using elementary row or column operations together with in-duction [95, 117] but there are also several proofs using combinatorial [11] or graph-based techniques [60, 141]. Here we will present a simple proof, that dates back to some very early results on determinants [24] and has an interesting connection to the general concept of a determinant.
Proof of Theorem 1.2. There are many versions of this proof, see for
exam-ple [10, 21, 24, 71], with focus on different aspects of the proof. Here we will provide a fairly concise version that still makes all the steps of the proof clear. We start by only considering one of the variables xk, which gives a
single variable function vn(xk). From the general expression for the
deter-minant (Theorem 1.1) it is clear that vn(xk) must be a polynomial of degree
n in xk. We also know that if we let xk = xi for any 1≤ i ≤ n, i = k, the
determinant will be equal to zero since the corresponding matrix will have two identical columns. Thus if vn(xi) = 0 we can write
vn(xk) = P (xk) n i=1 i =k (xk− xi)
where P (xk) is a polynomial. If we repeat this argument for all the variables,
and ensure that no roots appear twice in the factorization, we get
vn(x1, . . . , xn) = Pn(x1, . . . , xn) n −1 i=1 (xn− xi) = Pn−1(x1, . . . , xn) n −2 i=1 (xn−1− xi) n −1 i=1 (xn− xi) = P0(x1, . . . , xn)(x2− x1)(x3− x2)(x3− x1)· · · n −1 i=1 (xn− xi)
and since this factorization has each xk appear as a root n times we can conclude that vn(x1, . . . , xn) = C det(Vn(x1, . . . , xn)) = 1≤i<j≤n (xj− xi)
where C is a constant. From Leibniz formula for the determinant, equation (2), we can see that the coefficient in front of any term of the form xn
k must
be 1, thus C = 1, which concludes the proof.
Theorem 1.3. There is a relationship between the exponents of the expanded
Vandermonde determinant and the indices in the expression for a general determinant, more specifically
n i=1 xi vn(x1, . . . , xn) = n i=1 xi 1≤i<j≤n (xj− xi) = σ∈Sn (−1)I(σ) n i=1 xiσ(i). (3)
Clearly replacing xji with xi,j in equation (3) gives equation (2).
Proof. We will prove this theorem by showing that replacing exponents with
indices will give a function that by Definition 1.2 is a determinant. In Definition 1.2 we interpreted the determinant as a function of the columns of the matrix, for the Vandermonde determinant this corresponds to a function of the xi since they define the columns. Here we will interpret each part of
Definition 1.2 as a statement about the xi and then show how it is implied
by the Vandermonde determinant.
• Alternating: The alternating property is easy to interpret in terms
of the xi since if xi = xj for some i = j then we have two identical
columns. Consider the product form of the Vandermonde determinant given in Theorem 1.2. Switching places between xiand xjwith i < j in
the Vandermonde determinant is equal to switching sign in all factors that contain either xi or xj as well as xk with i ≤ k ≤ j. There will
be j− i − 1 factors that contain xi and satisfy i < k≤ j and j − i − 1
factors that contain xj satisfy i≤ k < j and one factor (xi− xj). This
means that in total we will change sign in 2(j− i) − 1 factors which
means the sign of the whole product will change.
• Multilinearity: If we denote the left hand side in (3) with w w = n k=1 xk vn(x1, . . . , xn)
then multiplying the kth column by a scalar can be interpreted as follows M·,k→ aM·,k⇔ w = n i=1 xikci→ n i=1 axikci
and addition of columns as
M·,k→ M·,k+ N·,k⇔ w = n i=1 xikci→ n i=1 (xik+ yki)ci
and multilinearity follows immediately from this.
• det(I) = 1: For the identity matrix we have xi,j =
1 i = j
0 i = j
which for the expanded Vandermonde determinant corresponds to the transformation
xji →
1 i = j
0 i = j
when expanding the Vandermonde determinant we get
vn(x1, . . . , xn) = vn−1(x1, . . . , xn−1) n−1 i=1 (xn− xi) = xnn−1vn−1(x1, . . . , xn−1) + P (n) = xnn−1vn−2(x1, . . . , xn−2) n −2 i=1 (xn−1− xi) + P (n) = xn−1n xn−2n−1vn−2(x1, . . . , xn−2) + P (n, n− 1) = n k=1 xkk−1+ P (n, n− 1, . . . , 1)
and since this factorization has each xk appear as a root n times we can conclude that vn(x1, . . . , xn) = C det(Vn(x1, . . . , xn)) = 1≤i<j≤n (xj− xi)
where C is a constant. From Leibniz formula for the determinant, equation (2), we can see that the coefficient in front of any term of the form xn
k must
be 1, thus C = 1, which concludes the proof.
Theorem 1.3. There is a relationship between the exponents of the expanded
Vandermonde determinant and the indices in the expression for a general determinant, more specifically
n i=1 xi vn(x1, . . . , xn) = n i=1 xi 1≤i<j≤n (xj− xi) = σ∈Sn (−1)I(σ) n i=1 xiσ(i). (3)
Clearly replacing xji with xi,j in equation (3) gives equation (2).
Proof. We will prove this theorem by showing that replacing exponents with
indices will give a function that by Definition 1.2 is a determinant. In Definition 1.2 we interpreted the determinant as a function of the columns of the matrix, for the Vandermonde determinant this corresponds to a function of the xi since they define the columns. Here we will interpret each part of
Definition 1.2 as a statement about the xi and then show how it is implied
by the Vandermonde determinant.
• Alternating: The alternating property is easy to interpret in terms
of the xi since if xi = xj for some i = j then we have two identical
columns. Consider the product form of the Vandermonde determinant given in Theorem 1.2. Switching places between xiand xjwith i < j in
the Vandermonde determinant is equal to switching sign in all factors that contain either xi or xj as well as xk with i≤ k ≤ j. There will
be j− i − 1 factors that contain xi and satisfy i < k≤ j and j − i − 1
factors that contain xj satisfy i≤ k < j and one factor (xi− xj). This
means that in total we will change sign in 2(j− i) − 1 factors which
means the sign of the whole product will change.
• Multilinearity: If we denote the left hand side in (3) with w w = n k=1 xk vn(x1, . . . , xn)
then multiplying the kth column by a scalar can be interpreted as follows M·,k→ aM·,k ⇔ w = n i=1 xikci→ n i=1 axikci
and addition of columns as
M·,k→ M·,k+ N·,k⇔ w = n i=1 xikci→ n i=1 (xik+ yik)ci
and multilinearity follows immediately from this.
• det(I) = 1: For the identity matrix we have xi,j =
1 i = j
0 i = j
which for the expanded Vandermonde determinant corresponds to the transformation
xji →
1 i = j
0 i = j
when expanding the Vandermonde determinant we get
vn(x1, . . . , xn) = vn−1(x1, . . . , xn−1) n−1 i=1 (xn− xi) = xnn−1vn−1(x1, . . . , xn−1) + P (n) = xnn−1vn−2(x1, . . . , xn−2) n −2 i=1 (xn−1− xi) + P (n) = xn−1n xn−2n−1vn−2(x1, . . . , xn−2) + P (n, n− 1) = n k=1 xkk−1+ P (n, n− 1, . . . , 1)
