[Ismo v. Lindell] Methods for Electromagnetic Fiel(BookFi.org)

METHODS FORELECTROMAGNETIC

FIELD ANALYSIS

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BOOKS IN THE IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY

Christopoulos, C., The Transmission-Line Modeling Methods: TLMClemmow, P. C., The Plane Wave Spectrum Representation ofElectromagnetic FieldsCollin, R. E., Field TheoryofGuided Waves, Second EditionDudley, D. G., Mathematical FoundationsforElectromagnetic TheoryElliot, R. S., Electromagnetics: History, Theory, andApplicationsFelsen, L. B., and Marcuvitz, N., RadiationandScatteringofWavesHarrington, R. F., Field Computation by MomentMethodsHansen et al., Plane-Wave TheoryofTime-Domain Fields: Near-Field ScanningApplicationsJones,D. S., Methodsin Electromagnetic Wave Propagation, Second EditionLindell, I. V., Methodsfor Electromagnetic FieldAnalysisPeterson et al., Computational Methods for ElectromagneticsTai,c.T., GeneralizedVector and DyadicAnalysis: AppliedMathematics in Field TheoryTai, C. T., DyadicGreenFunctions in Electromagnetic Theory, Second EditionVanBladel,J., SingularElectromagnetic FieldsandSourcesVolakiset al., FiniteElementMethod/or Electromagnetics: Antennas, Microwave Circuits, and ScatteringApplicationsWait,J., Electromagnetic Waves in StratifiedMedia

METHODS FORELECTROMAGNETIC

FIELD ANALYSIS

!sMO V. LINDELL

HELsINKI UNIVERSITY OFTECHNOLOGY

IEEEPRESS

TheInstitute of ElectricalandElectronics Engineers, Inc., NewYork

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Ismo V. Lindell, 1992.

All rights reserved No part ofthis book may be reproduced in anyform,nor may it be stored in a retrieval system or transmitted in any form,without written permission from the publisher.

10 9 8 7 6 5 4 3 2

ISBN 0-7803-6039-7IEEE Order No. PP5625

Library of Congress Cataloging-in-Publication Data

Lindell, Ismo v.Methods for electromagnetic field analysis I Ismo V. Lindell.

p. em.- (IEEE Press series on electromagnetic wave theory)Originally published: New York: Oxford University Press, 1992.IEEE order no. PP5625-T.p. verso.Includes bibliographical references and index.ISBN 0-7803-6039-71. Electromagnetic fields-Mathematics. I. Title. II. Series.

QC665.FA IA6 2000530.14'1--dc21

00-047149

PrefaceThe present monograph discusses a number of mathematical and concep-tual methods applicable in the analysis of electromagnetic fields. Theleading tone is dyadic algebra. It is applied in the form originated byJ.W. Gibbs more than one hundred years ago, with new powerful iden-tities added, making coordinate-dependent operations in electromagneticsall but obsolete. The chapters on complex vectors and dyadics are indepen-dent of the rest of the book, actually independent of electromagnetics, sothey can be applied in other branches of physics as well. It is claimed thatby rnemorizing about five basic dyadic identities (similar to the well-knownbac-cab rule in vector algebra), a working knowledge of dyadic algebra isobtained. To save the memory, a collection of these basic dyadic identities,together with their most important special cases, is given as an appendix.In different chapters the dyadics are seen in action. It is shown how simplydifferent properties can be expressed in terms of dyadics: boundary andinterface conditions, medium equations, solving Green functions, generaliz-ing circuit theory to vector field problems with dyadic impedances, findingtransformations between field problems and, finally, working on multipoleand image sources for different problems.

Dyadic algebra is seen especially to aid in solving electromagnetic prob-lems involving different linear media. In recent years, the chiral mediumwith its wide range of potential applications has directed theoretical inter-est to new materials. The most general isotropic medium, the bi-isotropicrnedium, has made electromagnetic theory a fresh subject again, with newphenomena being looked for. The rnedium aspect is carried along in thistext. What is normally analysed in isotropic media is done here for bi-isotropic or sometimes for bianisotropic media, if possible. Especially newis the duality transformation, which actually exists as a pair of transforma-tions. It is seen to shed new light on the plus and minus field decomposition,which has proved useful for analysing fields in chiral media, by showing thatthey are nothing more than self-dual fields with respect to each of the twotransformations.

In Chapter 5, Green dyadics for different kinds of media are discussedand a systematic method for their solution, without applying the Fouriertransformation, is given. In Chapter 6, source equivalence and its rela-tion to non-radiating sources is discussed, together with certain equivalent

vi PREFACE

sources: point sources (multipoles) and surface sources (Huygens' sources).Everywhere in the text the main emphasis is not on specific results butmethods of analysis.

The final chapter gives a summary of the work done by this authorand colleagues on the EIT, exact image theory. This is a general methodfor solving problems involving layered media by replacing them by imagesources which are located in complex space. The EIT is presented here forthe first time in book form.

The contents of this monograph reflect some of the work done andcourses given by this author during the last two decades. The resultsshould be of interest to scientists doing research work in electromagnetics,as well as to graduate students. For classroom use, there are numerouspossibilities for homework problems requiring the student to fill in stepswhich have been omitted to keep the size of this monograph within certainlimits. The EIT can also be studied independently and additional material,not found in this text, exists in print (see referece lists at section ends ofChapter 7).

The text has been typed and figures drawn by the author alone, leavingno-one else to blame. On the other hand, during graduate courses givenon the material, many students have helped in checking a great number ofequations. Also, the material of Chapters 1 and 2 has been given earlieras a laboratory report and a few misprints have been pointed out by someinternational readers. For all these I am thankful. The rest of the errorsand misprints are still there to be found.

This book is dedicated to my wife Liisa. A wise man is recognized forhaving a wife wiser than himself. I have the pleasure to consider myself awise man.

HelsinkiJuly 1991.

LV.L.

For the second edition, the main text has remained unchanged exceptfor a small number of misprints, which have been corrected. To assist inclassroom use of the book, three new appendices have been added. Ap-pendix A contains a collection of problems on the topics of Chapters 1 - 6and Appendix B a set of solutions for most of these problems. AppendixC gives a collection of most useful formulas in vector analysis for conve-nience. Appendix A (Dyadic identities) of the first edition is now relabeledas Appendix D.

HelsinkiMarch 1995.

LV.L.

Contents1 Complex vectors

1.1 Notation1.2 Complex vector identities1.3 Parallel and perpendicular vectors1.4 Axial representation1.5 Polarization vectors1.6 Complex vector basesReferences

2 Dyadics2.1 Notation

2.1.1 Dyads and polyads2.1.2 Symmetric and antisymmetric dyadics

2.2 Dyadics as linear mappings2.3 Products of dyadics

2.3.1 Dot-product algebra2.3.2 Double-dot product2.3.3 Double-cross product

2.4 Invariants and inverses2.5 Solving dyadic equations

2.5.1 Linear equations2.5.2 Quadratic equations2.5.3 Shearers

2.6 The eigenvalue problem2.7 Hermitian and positive definite dyadics

2.7.1 Hermitian dyadics2.7.2 Positive definite dyadics

2.8 Special dyadics2.8.1 Rotation dyadics2.8.2 Reflection dyadics2.8.3 Uniaxial dyadics2.8.4 Gyrotropic dyadics

11458

101415

1717171920222224252830313335363939404141434445

viii

2.9 Two-dimensional dyadics2.9.1 Eigendyadics2.9.2 Base dyadics2.9.3 The inverse dyadic2.9.4 Dyadic square roots

References

3 Field equations3.1 The Maxwell equations

3.1.1 Operator equations3.1.2 Medium equations3.1.3 Wave equationsReferences

3.2 Fourier transformations3.2.1 Fourier transformation in time3.2.2 Fourier transformation in spaceReferences

3.3 Electromagnetic potentials3.3.1 Vector and scalar potentials3.3.2 The Hertz vector3.3.3 Scalar Hertz potentialsReferences

3.4 Boundary, interface and sheet conditions3.4.1 Discontinuities in fields, sources and media3.4.2 Boundary conditions3.4.3 Interface conditions3.4.4 Sheet conditions3.4.5 Boundary and sheet impedance operatorsReferences

3.5 Uniqueness3.5.1 Electrostatic problem3.5.2 Scalar electromagnetic problem3.5.3 Vector electromagnetic problemReferences

3.6 Conditions for medium parameters3.6.1 Energy conditions3.6.2 Reciprocity conditionsReferences

CONTENTS

474849505051

53535556575859596263636466676869697174778183848586888889899394

CONTENTS ix

4 Field transformations 974.1 Reversal transformations 97

4.1.1 Polarity reversal 974.1.2 Time reversal 984.1.3 Space inversion 994.1.4 Transformations of power and impedance 99References 99

4.2 Duality transformations 1004.2.1 Simple duality 1004.2.2 Duality transformations for isotropic media 1004.2.3 Left-hand and right-hand transformations 1014.2.4 Application of the duality transformations 1034.2.5 Self-dual problems 1054.2.6 Self-dual field decomposition 1074.2.7 Duality transformations for bi-isotropic media 110References 112

4.3 Affine transformations 1124.3.1 Transformation of fields and sources 1134.3.2 Transformation of media 1154.3.3 Involutory affine transformations 116

4.4 Reflection transformations 1174.4.1 Invariance of media 1174.4.2 Electric and magnetic reflections 1184.4.3 The mirror image principle 1194.4.4 Images in parallel planes 1214.4.5 Babinet's principle 122References 124

5 Electromagnetic field solutions 1255.1 The Green function 125

5.1.1 Green dyadics of polynomial operators 1265.1.2 Examples of operators 128References 129

5.2 Green functions for homogeneous media 1295.2.1 Isotropic medium 1295.2.2 Bi-isotropic medium 1325.2.3 Anisotropic medium 133References 137

x CONTENTS

5.3 Special Green functions 1385.3.1 Two-dimensional Green function 1385.3.2 One-dimensional Green function 1405.3.3 Half-space Green function 141References 141

5.4 Singularity of the Green dyadic 1425.4.1 Constant volume current 1425.4.2 Constant planar current sheet 1445.4.3 Singularity for a volume source 1465.4.4 Singularity for a surface source 147References 148

5.5 Complex source point Green function 1485.5.1 Complex distance function 1495.5.2 Point source in complex space 1505.5.3 Green function 151References 153

5.6 Plane waves 1545.6.1 Dispersion equations 1545.6.2 Isotropic medium 1565.6.3 Bi-isotropic medium 1565.6.4 Anisotropic medium 157References 163

6 Source equivalence 1656.1 Non-radiating sources 165

6.1.1 Electric sources in isotropic medium 1656.1.2 Sources in bianisotropic media 168References 169

6.2 Equivalent electric and magnetic sources 1706.2.1 Bianisotropic medium 1706.2.2 Isotropic medium 1716.2.3 Chiral medium 173References 174

6.3 Multipole sources 1746.3.1 Delta expansions 1756.3.2 Multipole expansion 1766.3.3 Dipole approximation of a multipole source 1786.3.4 Electric and magnetic dipole approximation 181References 181

CONTENTS xi

6.4 Huygens' sources 1826.4.1 The truncated problem 1826.4.2 Huygens' principle 1836.4.3 Consequences of Huygens' principle 1856.4.4 Surface integral equations 1876.4.5 Integral equations for bodies with impedance surface 1896.4.6 Integral equations for material bodies 191References 191

6.5 TE/TM decomposition of sources 1926.5.1 Decomposition identity 1936.5.2 Line source decomposition 1946.5.3 Plane source decomposition 1966.5.4 Point source decomposition 197References 197

7 Exact image theory 1997.1 General formulation for layered media 199

7.1.1 Fourier transformations 1997.1.2 Image functions for reflection fields 2057.1.3 Image functions for transmission fields 2147.1.4 Green functions 218References 223

7.2 Surface problems 2247.2.1 Impedance surface 2247.2.2 Impedance sheet 2287.2.3 Wire grid 231References 235

7.3 The Sommerfeld half-space problem 2357.3.1 Reflection coefficients 2367.3.2 Reflection image functions 2387.3.3 Antenna above the ground 2477.3.4 Transmission coefficients 2497.3.5 Transmission image functions 2507.3.6 Radiation from a loop antenna into the ground 2557.3.7 Scattering from an object in front of an interface 257References 259

7.4 Microstrip geometry 2607.4.1 Reflection coefficients and image functions 2607.4.2 Fields at the interface 267

xii

7.4.3 The guided modes7.4.4 Properties of the Green functionsReferences

7.5 Anisotropic half spaceReferences

Appendix A Problems

Appendix B Solutions

Appendix C Vector formulas

Appendix D Dyadic identities

Author index

Subject index

CONTENTS

271272274275279

281

291

307

311

315

317

Chapter 1

Complex vectorsComplex vectors are vectors whose components can be complex numbers.They were introduced by the famous American physicist J. WILLARDGIBBS, sometimes called the 'Maxwell of America', at about the sameperiod in the 1880's as the real vector algebra, in a privately printed butwidely circulated pamphlet Elements of vector analysis. Gibbs called thesecomplex extensions of vectors 'bivectors' and they were needed, for ex-ample, in his analysis of time-harmonic optical fields in crystals. In alater book compiled by Gibbs's student WILSON in 1909, the text reap-peared in extended form, but with only few new ideas (GIBBS and WILSON1909). Thenceforth, complex vectors have been treated mainly in bookson electromagnetics in the context of time-harmonic fields. Instead of afull application of complex vector algebra, the analyses, however, mostlymade use of trigonometric function calculations. As will be seen in thischapter, complex vector algebra offers a simple method for the analysis oftime-harmonic fields. In fact, it is possible to use many of the rules knownfrom real vector algebra, although not all the conclusions. Properties ofthe ellipse of time-harmonic vectors can be seen to be directly obtainablethrough operations on complex vectors.

1.1 Notation

As mentioned above, complex vector formalism is applied in electromag-netics when dealing with time-harmonic field quantities. A time-harmonicfield vector F( t), or 'sinusoidal field' is any real vector function of time tthat satisfies the differential equation

(1.1)

A general solution can be expressed in terms of two constant real vectorsF 1 and F 2 in the form

F(t) = F 1 coswt + F 2 sinwt. (1.2)

2 CHAPTER 1. COMPLEX VECTORS

The complex vector formalism can be used to replace the time-harmonicvectors provided the angular frequency w is constant. There are certainadvantages to this change in notation and, of course, the disadvantagethat some new concepts and formulas must be learned. The main bulkof formulas, however, is the same as for real vectors. As an advantage,in using complex vector algebra, work with trigonometric formulas can beavoided, and the formulas look much simpler.

The complex vector f is defined as a combination of two real vectors,fre the real part, and fim the imaginary part of f:

(1.3)The subscripts re and im can be conceived as operators, giving the realand, respectively, the imaginary parts of a complex vector.

The essential point in the complex vector formalism lies in the one-to-one correspondence with the time-harmonic vectors f f-+ F(t). In fact,there are two mappings which give a unique time-harmonic vector for agiven complex vector and vice versa. They are:

f ~ F(t): F(t) = ~{feiwt} = fre coswt - fim sinwt, (1.4)F(t) --+ f: f = F(O) - jF(1r/2w) = F 1 - jF2 (1.5)

Thus, for the two representations (1.2) and (1.3) we can see the correspon-dences fre = F 1 and fim = -F2 .

The mappings (1.4), (1.5) are each other's inverses, as is easy to show.For example, let us insert (1.4) into (1.5):

(1.6)

which results in the identity f = f.It is important to note that there always exists a time-harmonic coun-

terpart to a complex vector whatever its origin. In fact, in analysis, therearise complex vectors, which do not represent a time-harmonic field quan-tity, for example the wave vector k or the Poynting vector P. We can,however, always define a time-harmonic vector through (1.4), maybe lack-ing physical content but helpful in forming a mental picture.

A time-harmonic vector F(t) = F 1 coswt +F 2 sinwt traces an ellipse inspace, which may reduce to a line segment or a circle. This is seen fromthe following reasoning.

If F 1 x F 2 = 0, the vectors are parallel or at least one of them is anull vector. Hence, F(t) is either a null vector or moves along a lineand is called linearly polarized (LP).

1.1. NOTATION 3

If F l x F 2 =1= 0, the vectors define a plane, in which the vector F(t)rotates. Forming the auxiliary vectors b = F l X (Fl x F2) andc = F 2 X (F1 x F 2) , we can easily see that the equation [b- F(t))2 +(c F(t))2 = IFl X F 2 14 is satisfied. This is a second order equation,whose solution F(t) is obviously finite for all t, whence the curve ittraces is an ellipse.

The special case of a circularly polarized (CP) vector is obtained,when IF(t)12 =Fjcos2wt+F~sin2wt+FlF2sin2wt is constant forall t. Taking t = 0 and t = 1r/2w gives F~ = F~, which leads to thesecond condition F l F 2 = O.

Thus, to every complex vector f there corresponds an ellipse just as forevery real vector there corresponds an arrow in space. The real and imag-inary parts fre, fim both lie on the ellipse. fre equals the time origin valueand is called the phase vector of the ellipse. The direction of rotation ofF(t) on the ellipse equals that of fim turned towards fre in the shortest way.A complex vector which is not linearly polarized (NLP) has a handednessof rotation, which depends on the direction of aspect. The rotation is righthanded when looked at in a direction u (a real vector) such that fim X fre Uis a positive number and, conversely, left handed if it is negative.

An LP vector must be represented by a double-headed arrow (infinitelythin ellipse), which is in distinction with the one-headed arrow represen-tation of real vectors. The difference is of course due to the fact that thetime-harmonic vector (1.2) oscillates between its two extremities.

The complex conjugate of a complex vector f, denoted by f*, is definedby

(1.7)From (1.4) we can see that f* corresponds to the time-dependent vectorF(-t), or it rotates in the opposite direction along the same ellipse as f(t).

The complex vector f is LP if and only if frex fim = O. This is equivalentwith the following condition:

f is LP :} f x f* = O.

The corresponding condition for the CP vector is

f is CP :} f f = O.

(1.8)

(1.9)

In fact, (1.9) implies that f:e = f?m and fre fim = 0, which is equivalentwith the CP property of the corresponding time-harmonic vector, as wasseen above.


Every LP vector can be written as a multiple of a real unit vector uin the form f = au. Every CP vector f can be written in terms of twoorthogonal real unit vectors u, v in the form f = a(u + jv). In theseexpressions a is a complex scalar, in general.

1.2 Complex vector identities

The algebra of complex vectors obeys many of the rules known from the realvector algebra, but not all. For example, the implication a-a = 0 =} a = 0is not valid for complex vectors. To be more confident in using identitiesof real vector algebra, the following theorem appears useful:

all multilinear identities valid for real vectors are also valid forcomplex vectors.

A multilinear function F of vector arguments aI, a2, ... is a functionwhich is linear in every argument, or the following is valid for i = 1 ... n:

F(al' a2, ..., (aa~ + ,Ba~/), ..., an) =aF(al, ..., a~, ..., an) + ,BF(al' ..., a~/, ...an).

A multilinear identity is of the form

F(al' ..., an) = 0 for all ai, i = 1...n.

(1.10)

(1.11)

Now, if the identity is valid for real vectors a, and the function does notinvolve a conjugation operation, from the linearity property (1.10) we canshow that it must be valid for complex vectors a, as well. In fact, takingQ = 1,,B = i, the identity is obviously valid if the real vector a, is replacedby the complex vector a~+ja~/. This can be repeated for every i and, thus,all vectors a, can be complex in the identity (1.11). As an example of atrilinear identity we might write

a x (b x c) - (a c)b + (a b)c = 0 for all a, b, c. (1.12)Also, all non-linear identities which can be derived from multilinear

identities are valid for complex vectors, like a x a = 0 for all vectors a. Theconjugation operation can be introduced by inserting conjugated complexvectors in multilinear identities. Thus, the identity

can be obtained from the real quadrilinear identity

(a x b) (c x d) = (a c)(b d) - (a d)(b c),

(1.13)

(1.14)

1.3. PARALLEL AND PERPENDICULAR VECTORS 5

with the substitution c = a", d = b*. The absolute value for a complexvector is defined by

lal2 = a a". (1.15)All irnplications that can be derived from identities are valid for complex

vectors if they are valid for real vectors. The basis for these is the null vectorproperty:

[a] = 0 :> a = 0, (1.16)which can be shown to be valid by expanding lal2 = lare + jainll2.

Two important, although simple looking theorems can be obtained fromvector identities:

a x b = 0 and a =j:. 0 :::} 30, b = oa,

a . b = 0 and a =1= 0 :::} 3c, b = c x a.(1.17) follows from the identity

a" x (a x b) = (a* b)a - (a a*)b,

(1.17)

(1.18)

(1.19)

from which b can be solved. Correspondingly, (1.18) is obtained from

a x (a* x b) = (a b)a* - (a a*)b.

As a consequence, from (1.17), (1.18) we see that the theorem

a x b = 0 and a b = 0 :::} a = 0 or b = 0,

(1.20)

(1.21)

valid for real vectors, is not valid for complex vectors. In fact, assuming[a] =1= 0 gives us either b = 0 or a is CP, which implies b is CPo A theoremcorresponding to (1.21) for complex vectors is the following one:

a x b = 0 and a b* = 0 :::} a = 0 or b = 0,

as can be readily verified from (1.17), (1.18).

1.3 Parallel and perpendicular vectors

(1.22)

For real vectors a, b there exist the geometrical concepts of parallel vectorsfor a x b = 0 and perpendicularity for a . b = O. Although the geometricalcontent will be different, it is helpful to define parallelity and perpendicu-larity with these same equations for complex vectors. This leads, however,to the existence of vectors which are perpendicular to themselves, namelythe CP vectors.


A complex vector b is parallel to a non-null vector a if there exists acomplex scalar 0: such that b = oa, as implied by (1.17). Let us denotea = Aej 8 with real A,O, and A > O. It is easy to see from the definition(1.4) that if we have the correspondence

then we also have

a ~ A(t),

()~ AA(t+-).

w

(1.23)

(1.24)

Thus, the magnitude of the ellipse is multiplied by the factor A and thephase of the ellipse is shifted by ()/w. The form of the ellipse as well asits axial directions and sense of rotation are the same for parallel vectors.Parallel vectors are said to have the same polarization.

The geometrical content of perpendicular complex vectors is more dif-ficult to express. Let us find the most general vector b perpendicular toa given non-null vector a. Obviously, if a is LP, or parallel to a real unitvector u: a = au, b may be any vector in the plane perpendicular to u, orof the form u x c.

For an NLP vector a there exists a real unit vector n satisfying n- a = 0,which is normal to the ellipse of a. Writing b = ,Bn+ba with b, in the planeof R, we see that {3 may have an arbitrary complex value. The problemis to find b.,, which must also be perpendicular to a. From the identityb; x (n x a) = n(ba a) - a(ba . n), whose right-hand side vanishes, and(1.17) we see that there must exist a scalar a such that b., = an x a.

Thus, the most general vector b perpendicular to a can be written as

b = ,Bn+ an x a, with n a = o. (1.25)

The corresponding time-harmonic vector B(t) is easily seen to be a sumof an LP vector along n plus an NLP vector in the plane of a, which isobtained from a vector parallel to a rotated by 1r /2 in its plane. The mostgeneral b vector can be seen to lie on an elliptic cylinder, whose crosssection is the ellipse of b a . It is also easy to see that there exist vectorsorthogonal to a given vector a of any ellipticity, because we can obtainellipses with every axial ratio by cutting the elliptic cylinder with planesof different orientations. In particular, the LP vector b is a multiple of n,whereas the two CP vectors

b = a(n x ajya:an), (1.26)

1.3. PARALLEL AND PERPENDICULAR VECTORS

are obtained from (1.25) with the CP condition b b = o.

7

b

b

(a)8(0)

(b)

Fig. 1.1 (a) Parallel complex vectors a and b. (b) Elliptic cylinderconstruction of a vector b perpendicular to a given vector a. The projection

vector b; is parallel to n x a, where n is a real unit vector normal to a.

Any vector b can be written as

b = b aa _ a x (a x b).aa aa

(1.27)

(1.28)

This gives a decomposition of a vector b into parts parallel and perpendic-ular to a vector a, and it is used in real vector algebra. Although (1.27)is also valid for complex vectors, it fails when a is CPo A more practicaldecomposition theorem is the following one:

b = b a a" _ a x (a* x b),a : a* a a*

which splits the vector b into vectors parallel to a" and perpendicular toa. This decomposition has the property of power orthogonality of its parts.In fact, writing (1.28) b = beD + b er as respective terms, where beD isthe co-polarized part and b er the cross-polarized part of the vector b withrespect to the vector a, we can write

(1.29)The co-polarized component is thus parallel not to a but to a". The def-inition is needed, for example, in antenna theory when reception of anincoming wave with the field vector E is considered with the polarizationmatch factor

Ih El2 [h X E*12p(h, E) = (h. h*)(E . E*) = 1 - (h. h*)(E . E*) , (1.30)

which tells us how well the polarization of the incoming field can be receivedby an antenna with the effective length vector h. It is seen that only the


co-polarized component of E with respect to h contributes to the valueof p(h, E) and complete polarization match p(h, E) == 1 is obtained forh x E* == 0, or when hand E* are parallel vectors. On the other hand,there is a total mismatch p(h, E) == 0 for' perpendicular vectors h, E, orwhen the incoming field is cross polarized with respect to the antennavector h.

As an example of the polarization match factor, let us consider radarreflection from an orthogonal plane. The far field of an antenna has thesame polarization as its effective length vector h. Reflection from the sur-face does not change the polarization of the field (but its handedness ischanged!), whence the polarization of the field coming back to the antennais also that of h. Because the polarization match factor is independent ofthe magnitude of the field, it equals p(h, h) in this case. It is seen thatfor an LP antenna h x h" == 0 and p(h, h) == 1, or there is no polariza-tion mismatch between the antenna and the incoming field. On the otherhand, if h is CP, we have p(h, h) == 0, or there is complete mismatch. ACP radar does .not see reflections from an orthogonal plane, or other circu-larly symmetric obstacles, wheras an LP antenna receives the best possiblesignal.

1.4 Axial representation

Polarization properties of a complex vector such as an electromagnetic fieldare often needed. For example, given a complex vector, how can we deter-mine its axial directions and magnitudes? Usually, in books working firstwith complex vectors, the notation is suddenly changed to time dependentrepresentation and the quantities needed are obtained through trigonomet-ric function analysis. This is, however, unnecessary, because the same canbe written down in complex vector notation quite simply. The procedureis based on the following simple facts.

(i) The vector b == ejBa has the same ellipse as a for real 8.(ii) There exists () real such that b., . binI == O.(iii) If b., . binI == 0, b re and b im lie on the axes of the ellipse of b.

(i) was demonstrated above and (ii) defines an equation for 8, which ob-viously has solutions. (iii) can be shown to be true through a consider-ation of the corresponding time-harmonic vector B(t), because IB(t)12 ==

1.4. AXIAL REPRESENTATION 9

8 1m

Fig. 1.2 Construction of the axis vectors of a complex vector a.

(1.31)

We are now ready to write an axial decomposition for any NCP vector a,from which the axes of its ellipse can be obtained. Obviously, a CP vectordoes not have any preferred axes, so it can be excluded. For an NCP vector,the scalar va:a is non-zero and we can define another complex vector bthrough

ab = ,va:-a,--.va:a

The factor multiplying a is obviously of the form e- j8 , whence b is of theform (i) above implying that b and a have the same axial vectors. There isno need to solve for B. Because b- b = [a-a] is real, (ii) is also satisfied, and(iii) is valid, whence the real and imaginary parts of b are the axis vectors.Since b . b is positive and equals b~e - b~m' the real part of the vectorlies on the major axis and the imaginary part of the vector on the minoraxis of the ellipse of b and, hence, a. The axial representation of complexvectors was probably first given by MULLER (1969) in his monograph onelectromagnetic theory.

The axial decomposition (1.31) giving the major axis vectors defined by

b re = 1~1!R{J:.a}'and the minor axis vectors by

bim = lva:-al~{ J:.a}'(1.32)

(1.33)

can easily be memorized and applied to simplify the analysis.For example, we can write a a* = b b* = Ibrel2 + Ibiml2 , to obtain

a geometrical interpretation for the magnitude lal = Va a* of a complexvector a, as the hypotenuse of the right triangle defined by the vectors b reand bim in Fig. 1.2.


1.5 Polarization vectors

As stated above, two parallel vectors have the same polarization. Thus, thepolarization of a vector a consists of all its properties that are not changedwhen multiplied by a complex scalar a. Because this operation changesthe magnitude and phase of the ellipse, the following are left as propertiesof polarization:

plane of the ellipse (can be defined by its normal vector n), direction of rotation on the plane (right hand in the direction n), e, the axial ratio of the ellipse, which defines its form,

axial directions on the plane of the ellipse (major axis along Ul, minoraxis along U2).

Because the complex vectors are defined by 3+3=6 real parametersand complex scalars by 2 real parameters, the definition of the polarizationconcept requires 4 real parameters. For example, the unit vector n takes 2parameters to define, the axial ratio e one, and the direction of the majoraxis Ul on the plane one more parameter (angle on the plane). The minoraxis direction is then obtained as U2 = n x uj ,

The polarization of a complex vector is very often of more interest thanthe complex vector itself. As one example, in certain microwave ferritedevices, a piece of ferrite material should be positioned in the spot wherethe magnetic field is circularly polarized, whatever the magnitude of thefield may be.

Polarization can be represented most naturally in terms of a normalizedcomplex vector U with

a=au. (1.34)For NCP vectors, a can be defined as ~, whence u is a complex unitvector satisfying u u = 1. For CP vectors, however, this breaks down. Asanother possibility we could try to define a as the real and positive number.;a:-a*, whence u is another complex unit vector satisfying u u" = 1, butthis u is no longer a representation of polarization because it contains thephase information of a.

p vector representation

A very useful way to present the polarization is by two real vectors p andq to be described next. p is defined as the following non-linear real vectorfunction of a:

a x a*p(a) = -.-.

Ja a*(1.35)

1.5. POLARIZArrION VECTORS

This vector has the following properties.

1. [p(a)]* = pea), or it is a real vector.

11

2. p(a*) == -p(a), or a change in the direction of rotation changes thedirection of p.

3. p(aa) == pea), or p is independent of the magnitude and phase of a.However, for a = 0 the p vector is indeterminate.

4. Ip(a)1 == 2e/(e2+1), where e is the ellipticity (axial ratio) of a. Hence,pea) == 0 {:} a is LP and Ip(a)( == 1 {:} a is CPo Otherwise thelength of p is between 0 and 1.

5. pea) = 2ainl X are/a a", whence for NLP vectors pea) points in thepositive normal direction of the a ellipse. The rotation of a is righthanded when looking in the direction pea).

6. p(acosO + n x asinO) = pea) for n == p(a)/lp(a)1 and 0 real. Thismakes sense for NLP vectors only and means that the ellipse maybe rotated in its plane by any angle () without changing its p vector.Thus, it is not sufficient to represent the polarization by the p vectoronly.

Although the real vector function pea) does not carryall the polariza-tion information of a, it is useful in analysing elliptic polarizations. In fact,it gives us the following information about a:

whether it is an LP vector or not;

for NLP vectors, the plane of polarization, sense of rotation and el-lipticity.

It does not give the following information:

direction, magnitude or phase of an LP vector;

for NLP vectors, the magnitude or phase or axial directions on theplane of polarization.

It is seen that the p vector provides least information for LP vectors andmost information for CP vectors, for which in fact the polarization is totally

12

known.

(a)

CHAPTER 1. COMPLEX VECTORS

p(a)

~-ta

.q;E >~a)(b) a

Fig. 1.3 Polarization vectors p(a) and q(a) of a complex vector a.

q vector representation

The complementary representation is the q(a) vector defined as follows:

q(a) = la al ~{a/va:a}a a* 1~{a/va:a}I (1.36)

In fact, (1.36) defines a pair of real vectors because of the two branchesof the square-root function. Hence, we may depict it as a double-headedarrow. The vector function q(a) has the following properties.

1. [q(a)]* = q(a), or it is a real function.2. q(a*) =: q(a), or the sense of rotation has no effect on q.3. q(aa) = q(a), or the magnitude and phase of a have no effect on the

q vector. For a null vector q is indeterminate.

4. Iq(a)1 = (1 - e2)/(1 + e2 ) , whence p2 + q2 = 1 and p, q is a com-plementary pair of vectors. q(a) --+ 0 for a approaching CP andIq(a)1 = 1 for a LP. Otherwise, the magnitude of q is between and1.

5. Ifa = ei 8b with b-b > 0, q(a) = brelq(a)I/lbrel, or q(a) is directedalong the major axis of a.

6. For a =: aUt + j(3u2 with real a, (3, uj , U2 and lit . U2 = 0, we haveq(a) = lq(a)IUt, or the direction of the minor axis of the ellipsedoes not affect on the q vector.

1.5. POLARIZATION VECTORS 13

Unit vectorrepresentationBecause real vectors have three parameters, it is not possible to representpolarization, requiring four parameters, by either of the p and q vectorsalone. Some information (like the ellipticity of the complex vector) is sharedby both vectors, in other respects they are complementary. It is possibleto form a pair of real unit vectors as a combination of the two real vectors:

u(a) = p(a) q(a), (1.37)which together exactly represent the polarization of the complex vector a.The subscript + or - is not essential, because the pair q is not ordered.The properties of u(a) can be listed as follows.

For a LP we have u., = -u+. Thus, a pair of opposite unit vectorsgives us the polarization of the LP vector.

For a CP, u., = u+. Coinciding unit vectors give the plane and senseof polarization of the CP vector.

In the general case, there is an angle 1/J between the unit vectors.From u., + u., = 2p(a) the plane and sense of polarization as well asthe ellipticity are obtained, whereas u., - u_+ = 2q(a) gives us thedirection of the major axis. The ellipticity and the angle 1/J have therelation e = cos(1/J/2)/(1 + sin'l/;/2)) = tan[(1r -1/J)/4].

(8) u 2 u 1( )

-q(a)

-q(a)

a

a

q(a)

q(a)

a

(c)

Fig. 1.4 Unit vector pair ui = u+(a), U2 = u-(a) representation of thecomplex vector a in (a) linearly polarized, (b) elliptically polarized and (c)

circular ly polarized cases.

In fact, any NCP complex vector can be written in the form

(1.38)

14

or in the equivalent form

CHAPTER 1. COMPLEX VECTORS

where

a = 20[(1 + Iq(a)l)q(a) + jq(a) x p(a)], (1.39)

aaa = ((1 + Iql)2 _ p2) q2 (1.40)

These expressions are not valid for CP vectors a, for which u., = u., andq = O. Equation (1.39) can, however, be extended to CP vectors if we letq ~ 0 so that oq = c is finite, whence

a = c + jc x p. (1.41)If only vectors a on a certain plane orthogonal to n are considered,

the direction of p(a) is fixed on the line n. Then, one single unit vectoris enough to represent the polarization of a, since the other one can beobtained from u., + u., = 2p. The unit vector pair corresponds to twopoints on a unit sphere. This is closely related to the well-known Poincaresphere representation of plane polarized vectors, where only a single pointis used. In fact, if a direction on.the plane is chosen, from which the angle

1.6. COMPLEX VECTOR BASES 15

(c X d) in two ways and equating the results. The vector triple a, b, c iscalled a base if a . b x c # 0, in which case (1.42) gives the decompositiontheorem

d = d a'a +d b'b +d c'c, (1.43)with the reciprocal base vectors defined by a' == b x c] J, h' = ex a/J, c' =a x hiJ, with J = a x b . c.

As an example of longitudinal ionospheric propagation in the directionu (real unit vector), a base corresponding to characteristic polarizationscan be formed with two CP vectors a, a* both orthogonal to u, satisfyingu x a == ja and u x a" == -ja*. Because p(a) == p(u x a) = u, a hasright-hand and a" left-hand polarization with respect to the direction ofpropagation u. The vector triple is a base, since u . a x a" == ja . a" f:. O.Hence, any field vector E can be expanded as

a* E * a EE == a--* +a --* +uuE.

aa aa(1.44)

A similar base can be generated from any NLP vector a as the triplea, a", p(a), because obviously a x a" . p(a) == j(a a*)p(a) . p(a), whichis nonzero for p(a) # 0.

An interesting and natural base can be generated from any NCP vectora through the following eigenvalue problem:

a x v = Xv, (1.45)

The eigenvalue A can be easily seen to have the values 0, jva:a., -jva:a.corresponding to the respective eigenvectors v = a, v +, v _, defined by

(.a x p(a))

V = O:' pea) =fJ va:a. ' (1.46)

with arbitrary coefficients o. The vectors defined by (1.46) are easily seento be CP vectors. If a is CP, the base does not exist. In fact, all threevectors tend to the same vector a as it approaches circular polarization.Also, for LP vectors a (1.46) does not seem to work because p(a) = o.However, the limit exists as a approaches linear polarization, if the productap(a) is kept finite.

References

DESCHAMPS, G.A. (1951). Geometrical representation of the polarizationof a plane electromagnetic wave. Proceedings of the IRE, 39, (5), 540-4.


DESCHAMPS, G.A. (1972). Complex vectors in electromagnetics. Un-published lecture notes, University of Illinois, Urbana, IL.

GIBBS, J.W. (1881,1884). Elements of vector analysis. Privatelyprinted in two parts, 1881 and 1884, New Haven. Reprint in The scientificpapers of J. Willard Gibbs, vol. 2, pp. 84-90, Dover, New York, 1961.

GIBBS, J.W. and WILSON, E.B. (1909). Vector Analysis, pp. 426-36.Scribner, New York. Reprint, Dover, New York, 1960.

LINDELL, I.V. (1983). Complex vector algebra in electromagnetics.International Journal of Electrical Engineering Education, 20, (1), 33-47.

MULLER, C. (1969). Foundations of the mathematical theory of elec-tromagnetic waves, pp. 339-41. Springer, New York.

Chapter 2

DyadicsDyadics are linear functions of vectors. In real vector space they can bevisualized through their operation on vectors, which for real vectors con-sists of turning and stretching the vector arrow. In complex vector spacethey correspondingly rotate and deform ellipses. Dyadic notation was in-troduced by GIBBS in the same pamphlet as the original vector algebra, in1884, containing 30 pages of basic operations on dyadics. Double productsof dyadics, which give the notation much of its power, were introduced byhim in scientific journals (GIBBS 1886, 1891). Gibbs's work on dyadic alge-bra was compiled from his lectures by WILSON and printed a book Vectoranalysis containing 150 pages of dyadics (GIBBS and WILSON 1909). Ofcourse, not all the formulas given by Gibbs were invented by Gibbs, quitea number of properties of linear vector functions were introduced earlierby Hamilton in his famous book on quaternions. In electromagnetics liter-ature, dyadics and matrices are often used simultaneously. It is well rec-ognized that the dyadic notation is best matched to the vector notation.Nevertheless, often the vector notation is suddenly changed to matrices,for example when inverse dyadics should be constructed, because the cor-responding dyadic operations are unknown. The purpose of this section isto introduce the dyadic formalism, and subsequent chapters demonstratesome of its power. The contents of the present chapter are largely basedon work given earlier by this author in report form (LINDELL 1968, 1973a,1981).

2.1 Notation

2.1.1 Dyads and polyads

The dyadic product of two vectors a, b (complex in general) is denotedwithout any multiplication sign by ab and the result is called a dyad. Theorder of dyadic multiplication is essential, ab is in general different frombat

A polyad is a string of vectors multiplying each other by dyadic productsand denoted by ala2a3 ...an' For n = 1 we have a vector, n = 2 a dyad,

18 CHAPTER 2. DYADICS

n = 3 a triad and, in general, an n-ad. Polyads of the same rank ngenerate a linear space, whose mernbers are polynomials of n-ads. Thus, allpolynomials of dyads, or dyadic polynomials, or in short dyadics, are of theform a, b l +a2b 2 +... + ak b k . Similarly, n-adic polynomials form a linearspace of n-adics. A sum of two n-adics is an n-adic._Here we concentrateon the case n = 2, or dyadics, which are denoted by A, B, C, etc.

Dyadics and other polyadics arise in a natural manner in expressionsof vector algebra, where a linear operator is separated from the quantitythat is being operated upon. For exarnple, projection of a vector a onto aline which has the direction of the unit vector u can be written as u(u a).Here, the vectors u represent the operation on the vector a. Separatingthese from each other by moving the brackets of vector notation, gives riseto the dyad uu in the expression u(u . a) = (uu) . a.

A dyad is bilinear in its vector multiplicants:

(alaI + (}2 a 2)b = (}1(aIb) + (}2(a2b),

a(,81b l + ,82b2) == ,81(ab1 ) + ,82(ab2).

(2.1)(2.2)

This means that the same dyad or dyadic can be written in infinitely manydifferent polynomial forms, just like a vector can be written as a sum ofdifferent vectors. Whether two forms in fact represent the same dyadic(the same element in dyadic space), can be asserted if one of them can beobtained fro~the other through these bilinear operations.

A dyadic A can be multiplied by a vector c in many ways 0 Taking onedyad ab of the dyadic, the following multiplications are possible:

Co (ab) == (c vajb,c x (ab) == (c x a)b,(ab)oc==a(boc),

(ab) x c == a(b x c).

(2.3)(2.4)(2.5)(2.6)

In dot multiplication of a dyad by a vector, the result is a vector, in crossmultiplication, a dyad.

Likewise, double multiplications of a dyad ab by another dyad cd aredefined as follows:

(ab) : (cd) == (a c)(b d),(ab)~(cd) == (a x c)(b x d),(ab}" (cd) == (a x c)(b d),(ab); (cd) == (a c)(b x d).

(2.7)(2.8)(2.9)

(2.10)

2.1. NOTATION 19

These express~ns~a~b~g~eralized.!9corresponding double productsbetween dyadics, A : B, A~B, A~B and A~B, when dyads are replaced bydyadic polynomials and multiplication is made term by term. The doubledot product produces a scalar, the double cross product, a dyadic, andthe mixed products, a vector. These products, especially the double dotand double cross products, give more power to the dyadic notation. Theirapplication requires, however, a knowledge of some identities, which arenot in common use in the literature. These identities will be introducedlater and they are also listed in Appendix A of this book.

The linear space of dyadics contains all polynomials of dyads as its ele-ments. The representation of a dyadic by a dyadic polynomial is, however,not unique. Two polynomials correspond to the same dyadic if their differ-ence can be reduced to the null dyadic by bilinear operations. Because ofGibbs' identity (1.42), any dyadic can be written as a sum of three dyads.In fact, taking three base vectors a, b, c with their reciprocal base vectorsa', b', c', any dyadic polynomial can be written as

n n n nL a.b, == a L(a' . ai)b i + b L(b' . ai)bi + c L(c' .ai)bi. (2.11)i=l i=l i=l i=l

This is of the trinomial form ae + bf + cg, which is the most general formof dyadic in the three-dimensional vector space. If we can prove a theoremfor the general dyadic trinomial, the theorem is valid for any dyadic. Asum sign L without index limit values in this text denotes a sum from 1to 3.

2.1.2 Symmetric and antisymmetric dyadics

The transpose operation for dyadics changes the order in all dyadic prod-ucts:

(2.12)

Because (AT)T == A, the eigenvalue pr~lem AT == ,\A has the~igenvalues,\ == 1 corresponding to symmetric As and antisymmetric Aa dyadics,which satisfy

A; == As, AaT == -Aa. (2.13)Any dyadic can be uniquely decomposed into a symmetric and an antisym-metric part:

= 1= =T 1= =TA == 2(A + A ) + 2(A - A ). (2.14)Every symmetric dyadic can be written as a polynomial of symmetric

dyads:


= 1As = Laibi = "2 L(aibi + b.a.) =

1"2 L ((ai + bi)(ai'+ b.) - a.a, - b.b.}, (2.15)

The number of terms in this polynomial is, however, in general higher than3.

A single dyad cannot be antisymmetric, because from the conditionab == -ba, by multiplying by a* and dividing by a a" we see that bmust be of the form oa, whence ab = oaa == -aaa == O. Instead, anyantisymmetric dyadic can be expressed in terms of two dyads in the formab - ba. The vectors a, b are not unique. This is seen from the followingexpansion with orthonormal unit vectors Uj:

(2.16)

with(2.17)

(2.18)

Thus, the general antisymmetric dyadic can be expressed in terms of asingle vector c. If we write c == -a x b, going (2.16) backwards we seethat any antisymmetric dyadic can be expressed as ab - ba. The choice oforthonormal basis vectors does not affect this conclusion.

The linear space of dyadics is nine dimensional, in which the antisym-metric dyadics form a three-dimensional and the symmetric dyadics a six-dimensional subspace.

2.2 Dyadics as linear mappings

A dyadic serves as a linear mapping from a vector to another: a -+ b ==D . a. Conversely, any such linear mapping can be expressed in terms ofa dyadic. This can be seen by expanding in terms of orthonormal basisvectors u, and applying the property of linearity of the vector functionrea):

rea) = I: u.u, . f(I: UjUj . a) = (I: I: u, . f(Uj)UiUj) . a.j i

2.2. DYADICS AS LINEAR MAPPINGS 21

The quantity in brackets is of the dyadic form and it corresponds to thelinear function f(a). _ _

The unit dyadic 7 corresponds to the identity mapping 7 a = a forany vector a. From Gibbs' identity (1.42) we see that for any base of threevectors a, b, c with the reciprocal base a', b', e', the unit dyadic can bewritten as

1 = aa' + bb' + ee'. (2.19)Taking an orthonormal base u, with u~ = u., the unit dyadic takes theform

(2.20)

The unit dyadic is symmetric and satisfies 1 D = Dl = D for any dyadicD. This and (2.19) can be applied to demonstrate the relation betweenmatrix and dyadic notations by writing

(2.21)

or any dyadic can be written in terms of nine scalars D i j . These scalarscan be conceived as matrix components of the dyadic w~h respect to thebase {ail. The matrix components of the unit dyadic 1 are {6i j } in allbases.

From (2.16) it can be seen that the most general antisymmetric dyadiccan be written as

Aa ~ ab - ba = (b x a) x I = I x (b x a), (2.22)as is_seen_if (2.20) is substituted in (2.22). Thus, dyadics of the forme x 1 = 1 x e are antisymmetric. Th~vector e corresponding to theantisymmetric part of a general dyadic D can be obtained through thefollowing operation:

- 1--c(D) = 2.1~D. (2.23)

For an antisymmetric dyadic, (2.23) can be easily verified from (2.22). Fora symmetric dyadic, e(D) = 0 is also easily shown to be valid.

All dyadics can be classified in terms of their mapping properties.

Complete dyadics D define a linear mapping with an inverse, whichis represented by an inverse dyadic D-l. Thus, any vector b can bereached by mapping a suitable vector a by D a = b.

22 CHAP1'ER 2. DYADICS

Planar dyadics map all vectors in a two-dimensional subspace. If wetake a base [a.}, the vectors {D . a.] do not form a base, becausethey are linearly dependent and satisfy (Dal)(Da2) x (Da3) == o.Writing the general D in the trinomial form ab + cd + ef, we canshow that the vector triple a, c, e is linearly dependent and one ofthese vectors can be expressed in terms of other two. Thus, the mostgeneral planar dyadic can be written as a dyadic binomial ab + cd.

Strictly planar dyadics are planar dyadics which cannot be written asa single dyad.

Linear dyadics map a!!- vectors in a one-dimensional subspace, l.e.parallel to a vector c: D a == oc. Thus, D must obviously be of theform cb. Linear dyadics can be written as a single dyad. Finally, wecan distinguish between strictly linear dyadics and the null dyadic.

As examples, we note that the unit dyadic I is complete, whereas anantisymmetric dyadic is either strictly planar or the null dyadic. The in-verse dyadic of a given complete dyadic D == E a.b, can be written quitestraightforwardly in trinomial form. First, to be complete, the vector triples[a.}, {bi} must be bases because from linear dependence of either base, aplanar dyadic would result. Hence, there exist reciprocal bases {a~}, {b~},with which we can write

(2.24)

That (2.24) satisfies D . n-1 == D- 1 . D == I, can be easily verified.

2.3 Products of dyadics

Different products of dyadics playa role similar to dot and cross productsof vectors, which introduce the operational power to the vector notation.The products of dyadics obey certain rules which are governed by certainidentities summarized in Appendix A.

2.3.1 Dot-product algebra

The dot product between two dyadics has already been mentioned aboveand is defined in an obvious manner:

(2.25)

2.3. PRODUCTS OF DYADICS 23

With this dot product, the dyadics form an algebra, where the unit dyadic,null dyadic and inverse dyadics are defined as above. This algebra corre-sponds to the matrix algebra, because the matrix (with respect to a givenbase) of A . B can be shown to equal the matrix product of the matricesof each dyadic. Thus properties known from matrix algebra are valid todot-product algebra: associativity

A. (B C) = (A. B) C, (2.26)and (in general) non-commutativity, A B =1= B . A. Further, we have

(A . B)T = BT .AT,(A. B)-1 = B-1. A-I.

(2.27)

(2.28)Powers of dyadics, both positive and negative, are defined through the dotproduct (negative powers only for comp~te dyad~cs) in an obvious manner.For example, the antisymmetric dyadic A = u x 7 with an NCP unit vectoru, satisfies for all n > o.

A4n =1- uu,=A4n +l - A

- ,

A4n +2 - -1+ uu- ,

(2.29)(2.30)(2.31)(2.32)

Because for re~l u,. A can be interpreted as a rotation by 1r/2 around u,the powers of A can be easily understood as multiples of that rotation.

Two dyadics do not commute in general in the dot product. It is easy tosee that two antisymmetric dyadics only commute when one can be writtenas a multiple of the other. This is evident if we expand the dot product oftwo general antisymmetric dyadics:

(a x 1) . (b x 1) = ba - (a b)1. (2.33)If this is r~quired to be symmetric in a and b, we should have ab = ba or(a x b) x I = 0, which implies a x b = 0 or a and b are parallel vectors.

A dyadic commutes with an antisymmetric dyadic only if its symmetricand ~ntisymmetricparts commute separately. In fact, writing D = Ds +d x I in terms of its symmetric and antisymmetric parts, we can write

D . (a x I) - (a x I) .D = (D s x a) + (Ds x a)T - (a x d) x I. (2.34)


Equating the antisymmetric and symmetric parts !.(2.34) to zero, showsus t~at the symmetric and antisymmetric parts of D must commute witha x J. separately. Thus, the antisymmetric part of D must be a multiple ofa x 1. The symmetr~part of D must be such that D 8 X a is antisymmetric,i.e. of the form b x 1. Multiplying this by -a gives zero, whence b must bea multiple ~f 8. It is easy to show that the symmetric dyadic must be ofthe form 0./+ {3aa. Thus, the_most general dyadic, which commutes withthe antisymmetric dyadic 8 x I is necessarily of the form

D = QJ + {3aa+ ,a x I. (2.35)A dyadic of this special form is called gyrotropic with axis 8, which mayalso be a complex vector. From this, it is easy to show that if a dyadiccommutes with its transpose, it must be either symmetric or gyrotropic.

2.3.2 Double-dot product

The double-dot product of two dyadics A =L a.b., B =L cjdj gives thescalar

(2.36)This is symmetric in both dyadics and satisfies

(a x I) : (b x I) = 2a . b,A :I = 2)ai hi) = trA.

(2.37)

(2.38)(2.39)

The last operation gives a scalar which can be called the trace of A becauseit gives the trace of the matrix of A in any base [c.]. In fact, writing= , .A = L AijCiCj gives us

As special cases we have I :I = 3 and A : B = (If . BT) : I = (B . AT) : I.A dyadic whose trace is ze.!o is called trace free. Any dyadic can be writtenas a sum of a multiple of I and a trace-free dyadic:

- 1- -- - 1---D = -(D : 1)1+ (D - -(D : /)/).3 3 (2.41)

2.3. PRODUCTS OF DYADICS 25

Antisymmetric dyadics are trace free. In fact, more generally, if S issymmetric and A antisymmetric, ~e have from (2.37)

(2.42)

The scalar D : D* is a non-negative real number for any complex dyadicD. It is zero only for D = 0, which can be shown from the following:

= =* =T =* = ~ =r =* ~ = 2D : D = (D . D ): I == L...J u, . D . D . u, == L...J ID . u.] . (2.43)

Here, {Ui} is a real orthonormal base. (2.43) is seen t~give a non-negativenumber and vanish only if all D . u, vanish, whence D == L D . u.u, == O.We can define the norm of D as

IIDII = JD:D*. (2.44)

2.3.3 Double-cross product

The double-cross product of two dyadics produces a third dyadic. Thus, itdefines a double-cross algebra. Unlike the dot-product algebra, the double-cross algebra is commutative:

=x= =x=AxB = BxA, (2.45)

and non-associative, because A~(B~C) =/; (A~B)~C in general. The com-mutative property follows directly from the anticommutativity of the crossproduct: a x b == -b x a, as is easy to see. It is also easy to show thatthere does not exist a unit element in this algebra.

A most useful formula for the expansion of dyadic expressions can beobtained from the following evaluation with dyads:

(ab)~[(cd)~(ef)] = [a x (c x e)][b x (d x f)] ==[ca e - a ce][db . f - b df] =

(ab : cd)ef + (ab : ef)cd - ef (ab)T . cd - cd (ab)T . ef. (2.46)This expression is a trilinear identity for dyads. Thus, every dyad can bereplaced by any dyadic polynomial because of linearity, whence (2.46) maybe written for general dyadics:

(2.47)


Use of this dyadic identity adds more power to the dyadic calculus. Itsmemorizing is aided by the fact that, because of the commutative property(2.45), Band C are symmetrical in (2.47).

The method used above to obtain a dyadic identity from vector identi-ties can be generalized by the following procedure.

1. A multilinear dyadic expression (linear in every dyadic) is written interms of dyads, i.e. every dyadic is replaced by a dyad.

2. Vector identities are applied to change the expression into anotherform.

3. The result is grouped in such a way that the original dyads are formed.

4. The dyads are replaced by the original dyadics.

_ To demonstrate this procedure,~t us expand the dyadic expression(A~B) : I, which is linear in A and B. Hence, we start by replacing themby ab andcd, respectively, and applying the well-known vector identity

(ab~cd) : I = (a x c) (b x d) = (a b)(c d) - [a- d)(b c). This can begrouped as (ab : I)(cd : I) - (ab) : (cd)?'. Finally, going back to A and Bleaves us with the dyadic identity

== = ==== =nr(A~B) : I = (A : I)(B : I) - A : B , (2.48)or trace of A~B equals trA trB - tr(A B).

New dyadic identities can also be obtained from old identities. As anexample, let us write (2.48) in the form

(2.49)

To obtain this, we have applied the invariance in any permutation of the= = = = = = =x= =triple scalar product of dyadics, A~B : C = A~C : B = BxA : C =

... (Of course, the double-cross product must always be performed first.)Because (2.49) is valid for any dyadic B, the bracketed dyadic must be thenull dyadic, and the following identity is obtained:

=x= = == 7TAxI=(A:I)I-A. (2.50)

That D : B = 0 for all B implies D =0, is easily seen by taking B = UiUjfrom an orthonormal base [u.}, whence all matrix coefficients Dij of Dcan be shown to vanish. The identity (2.48) is obtained from (2.50) as aspecial case by operating it by : B.

2.3. PRODUCTS OF DYAD/CS 27

An important identity f~ the double-cross product can be obtained byexpanding the expression (A~B)~(C~D) twice through (2.47) by consid-ering one of the bracketed dy~dics as a single dyadic, and equating theexpressions. Setting C = D = / we obtain

=x=AxB=

This identity could be also conceived as the definition of the double-crossproduct in terms of single-dot and double-dot products and the transpose~p~atio~ It is easily seen that (2.50) is a special case of (2.5~. _ Also,/~1 = 2/ is obtained as a further special case. The operation A~/ is infact a mapping from dyadic to dyadic. Its properties can be examinedthrough the following dyadic eigenproblem:

=x= =Ax! = ..\A. (2.52)

Taking t~ trace of (2.52) leaves us with (2 - "\)A : / = 0, whence either,.\ = 2 or A is trace free. Substituting (2.50) in (2.52) gives us the followingdifferent solutions:

,.\ = 2 and A = a/ where a is any scalar;

,.\ = 1 and A is antisymmetric; ,.\ = -1 and it is symmetric and trace free.

Any dyajic can be written uniquely as a sum of three components: amultiple of 1, an antisymmetric dyadic and a trace-free symmetric dyadic,

(2.53)

respectively. It is a simple matter to check that the right-hand side of(2.53), each term multiplied by the corresponding eigen~alue ,.\, gives thesam~ ~sult as (2.50). T~r~e~sts a~ inverse maR.ping to 1~ which, denotedby I(A) and satisfying I(A~/) = A for every A, can be written in termsof inverse eigenvalues, or in the simple form

(2.54)


Finally, let us ~onsider the double-cross mapping ~hrough an antisym-metric dyadic d x 1. Every antisymmetric dyadic a x 1 is mapped onto theplanar symmetric dyadic

(a x 1) ~ (d x I) = ad + da. (2.55 )Every symmetric dyadic S is mapped onto the antisymmetric dyadic:

S~(dXI)=(S.d)XI. (2.56)The mapping of the general dyadic is obtained when it is decomposed intoa symmetric and an ~ntisymmetric part. There does not exist an i~verse tothe mapping ~ (d x I), because all symmetric dyadics satisfying S . d == 0are mapped onto the null dyadic.

2.4 Invariants and inverses

In matrix algebra, invariants are such functions of a matrix that are in-dependent of the basis in which the matrix is formed. Thus, they can beexpressed as functions ~ th! corresponding dyadic. The trace was definedin (2.39) as the scalar A : I, corresponding to the sum of diagonal termsin the matrix. The second invariant can be denoted, and expanded from(2.48), as

- 1- - - 1 - - - -spmA == -A~A: 1 = -[(A: 1)2A: AT],2 2 (2.57)

and its counterpart in matrix algebra is the sum of principal minors. Thethird invariant corresponds to the determinant of the matrix and can beexpressed as

- 1- - -detA = 6A~A; A.Finally, the cross-product square of the dyadic is defined as

A(2) = !A~A.2

(2.58)

(2.59)

As an example, 1(2) = t can be easily verified.Writing the dyadic A in the form Laibi, from (2.59) we can find the

following vector expression for the cross-product square:

(2.60)

2.4. INVARIANTS AND INVERSES 29

where {a'l and {hi} are reciprocal to the bases {a}, {b}. If A is not com-plete, the last expression in (2.60) is not meaningful, since the reciprocalbases do not both exist. Further, we can write for the determinant,

(2.61)

Combining (2.24), (2.60) and (2.61) gives us the formula for the inverse ofa dyadic:

which implies

= A(2)T (AXA)TA-I = ---=- = 3__x __- - - -,detA A~A: A

(2.62)

(2.63)

Because detI = 1, the in~rse of I is easily seen to be I, as expected. Theinverse exists only if detA ~ 0 is satisfied, which serves as a definition ofthe complete dyadic. In fact, the classification of dyadics can be written inthe form

complete dyadic if detA =f. o.

strictly planar dyadic if detA = 0 and A~A ~ o. linear dyadic if A~A = o.There exists a relation between the cross-product square of a dyadic

and its dot-product square. In fact, from (2.51) we can write

(2.64)

After transposing (2.64), dot-multiplying by if ~nd applying (2.62) we endup with a relation between powers of a dyadic if

(2.65)

which is called the Cayley-Hamilt~equation, but actually it is an identity,since it is satisfied by all dyadics A.

As other properties of the cross-product square we can easily prove that

(2.66)

30 CHAPTER 2. DYAD/CS

which tells ~t~t if A is planar, A~A is a linear dyadic. In fact, a dyadicof the form A~A cannot be st:!.ctly planaE! it is either complete or linear.

To prove the property det{An) = (detA)n known from matrix algebra,we need the identity

(2.67)which can be derived with the method described above. The following is aspecial case:

which can also be written as

A(2) . B(2) = (A . B)(2).Applying this and (2.63) we have

det{A B) = (detA){detB)

(2.68)

(2.69)

(2.70)after some algebra. This in fact shows that the dot product of any numberof dyadics has a determinant function which equals the product of deter-minant functions of individual dyadics. Thus, the product of dyadics isplanar if anyone of the dyadics is a planar dyadic.

2.5 Solving dyadic equations

In this section, some principles for and results of solving dyadic equationsare considered. The presentation does not cover all the kin~ of equations~at may occur. By a dyadic equation we mean the form A = B, whereA and B~ay ~ontain unkn~n quantities. In the following cases, theequation A = B reduces to A = 0 and B = 0, or is split up into twoequations which are usually easier to solve than the original one.

11 = aC where C is a complete dyadic, and B is known to be planar.(Taking the determinant function of each side gives a = 0.) If a =I 0,there is no solution at all.

A = 0:1 and B is trace free. (Taking the trace operation of each sidegives us a = 0.)

11 is symmetric and B is antisymmetric. A is antisymmetric and B is linear dyadic. (Because an antisymmetric

dyadic is either strictly planar or null dyadic, both are null dyadics.)

2.5. SOLVING DYADIC EQUATIONS

For example, we might encounter an equation of the type

=x= =AxA=axI.

31

(2.71)

Because A~A is either complete, linear or null dyadic and the antisymmet-ric dyadic a ~1 either strictly planar or null dyadic, we have a = 0 and

A~A = 0, or A may be any linear dyadic.

2.5.1 Linear equations

Let us study the linear dyadic equation

(2.72)

If A is complete, the solution X = A-i. B is defined by (2.62). Problemsarise when A is planar. If in this case B is complete, (2.72) does not possessa solution at all. Thus, for (2.72) to have a solution when 11 is planar, Bmust also be planar, which does not warrant a solution, however. If thereis a solution, it is not unique unless we restrict it somehow. Let u~studythe problem (2.72) where A is strictly planar, and try to solve for x.

Because A is strictly planar, we have detA = 0 and A(2) i= o. In fact,the cross-product square of A is a strictly linear dyadic and can be writtenin theJorm A(2) = ab =1= o. From (2.63) we have A(2)T . A = A A(2)T = 0,or a . A = A b = o. Thus, A defines a class A of planar dyadics which areorthogonal to a from the left and to b from the right. This class can beused to define a unique solution to the equation (2.72).

From (2.72) it is seen that there does not exist a solution unless Bsatisfies

A(2)T . B = 0, (2.73)

or a . B = O. B need not belong to class A. Let us look for the solution of(2.72) in the form X = DB and find a dyadic D, whose conjugate transposeis in class A. Such a dyadic is called the planar inverse or generalizedinverse of A and is simply denoted by D = A-1. Because the inverse inthe strict sense (2.62) does not exist for planar dyadics, there should notbe any place for misinterpretation due to the notation. The planar inversecan be written in the form

= (A X lI* (2)TA-1 = x .A(2) : A*(2)

(2.74)


This is a solution whose conjugated transpose is in class A, as can beeasily checked. To verify (2.74), we expand using (2.47) and (2.64):

A (A~(A*~A*))T= 2[tr(A. A*T)(A. A*T) - (A. A*T)2] =2[spm(A . A*T)I - (A . A*T)(2)T], (2.75)

whence we can write= (A~A*(2)T = (A* . AT)(2) =A __ =1- __ =lp (2.76)

spm(A* . AT) spm(A* . AT)Here, the dyadic I p can be~alled the planar unit dyadic of class A.

Multiplying (2.76) by B from the right and taking (2.69), (2.73) andthe identity

(2.77)into account, we see that X =_A-1 . B is really a solution of (2.72). (2.~)is called the planar inverse of A. The denominator in (2.74) equals spm(AA*T) and is never zero for strictly planar dyadics A.

The most general solution to (2.72) can be written in the formx = A-I. B + x ;

where X 0 is any solution to the homogeneous equation

AXo=O.

(2.78)

(2.79)Because ! is strictly planar, X 0 must be a linear dyadic, as is easily

seen writing A = al hI + a2 h 2 . In fact, we obviously must have= =(2)T =X; = A . C, (2.80)

as t.!:e most general solution with an arbitrary dyadic C. The equationX 11= B can be solved in corresponding manner.

Let us now study the linear equation

A~X = B, (2.81)

(2.82)

where A is a complete dyadic. Multiplying this by (A~ A) ~ and applying(2.47) we easily have for the solution

= =-lTx= A: B=X = A xB - ----:=A.2detA

For example, if A = I, the expression (2.54) is obtained.

2.5. SOLVING DYADIC EQUATIONS

2.5.2 Quadratic equationsLet us study the quadratic equation

33

(2.83)

which is quite easy to solve if A is a complete dyadic. Double-cross multi-plying both sides with itself and applying (2.66) gives us the two solutions

=x== AxAx= ~,

V8detA(2.84)

which fails for planar dyadics satisfying detA = o.If A is s~ictly planar, :4(2) t= 0, there is no solution at all, as was seen

earlier. If A is strictly linear, there exist infinitely many strictly planarsolutions X. In fact, in this case A defines a class of strictly planar dyadicssatisfying X .AT =AT .X = o. For A= 0, any linear dyadic is a solution.

Square roots of dyadics

The quadratic equation(2.85)

is more difficult to solve than (2.83). Its solutions, square roots of:4, maybe infinEe or zero in number, or something in be.!ween, depending on thedyadic A. For exa~ple, all dyadics of the form (1- 2uu) are square rootsof the unit dyadic 1 for all unit vectors u.

A solution procedure can be based on the solution of (2.83) through~e following identity, which can be derived by applying the identity forA~(B~C):

(2.86)

The procedure is first to solve for the unknown scalar a = spmX andthen solve (2.86). An equation for a can be derived as follows. DenotingD(o) = a1 + X 2 = a1 + A, we can write

(X~I)(2) = D(a),

[(X~I)(2)](2) = [det(X~I)]X~I = D(2),det[(X~I)(2)] = [det(X~I)]2 = detD.

(2.87)

(2.88)

(2.89)


(2.90)

For D(a) ~mplete, from (2.88) and (2.89) we can express the unknowndyadic x~I in terms of the unknown dyadic D(a):

_ _ D(2) D(2)X~I== __ == _,

det(X ~ I) JdetDwhence by taking the spm() operation of this and applying the identity(trX)2 = 2spmX + tr(X2) we obtain the following quartic equation for theunknown scalar a:

a 4 - 2a2spmA - 8a detA + (spmA)2 - 4trA detA == o.

Values of a substituted in (2.90) solved for X

X = 1 = ((spmD)I _ 2D(2)T)2VdetD

(2.91)

(2.92)

give us possible square root solutions corresponding to those cases for whichthe dyadic D(a) is complete. Written explicitly, the solution is

X = A1/ 2 = ~ _ ((spm(A + 0:1))1 - 2(A + o:I)(2)T) .2Vdet(A+ 0:1)(2.93)

If the completeness condition is not valid, there are either no or infinitelymany solutions corresponding to that particular solution of (2.83). As anexample of a dyadic with no square roots we may write

A=uv+ww, (2.94)where u, v, w is an orthon~rmal set of unit vectors. In fact, this cor-responds to detA = 0, spmA = 0, whence (2.91)~nly has the solutiona == o. Thus, 15 equals the strictly planar dyadic A, whence (2.88) doesnot possess a solution. A dyadic of this kind is called a shearer.

Square roots of the unit dyadic

As an example, le~us c..?nsider the problem of finding all the square rootsof the unit dyadic A == I. In this case, the quartic (2.91) can be written as

(a-3)(a+l)3=O,or the roots are a == 3 and a = -1. The equation (2.86) now reads

(2.95)

(2.96)

2.5. SOLVING DYADIC EQUATIONS 35

whose right side equals 41 and zero for the respective two a values. Thesolution formula (2.93) can n


Note that this defini!ion d~ffers from that adopted by Gibbs, who callsdyadics of the form A + Ctl shearers. From the Cayley-Hamilton iden-tity (2.65) we can write another condition equivalent to (2.102), for thedefinition of the shearer:

(2.103)It is easily seen that, in the trace-free case, (2.103) corresponds to (2.100).

The most general shearer can be written in the forms

A = a x (be +da), or A = (ad + eb) x a. (2.104)

It is not difficult to check that (2.102) and (2.103) are satisfied for anyvectors a, b, c, d. As was seen just before, a strictly planar shearer doesnot have a square root.

2.6 The eigenvalue problem

Right and left eigenvalue problems with eigenvalues and eigenvectors cq, a,and, respectively, {3i, hi, are of the form

(2.105)

(2.106)Because the dyadic A-,1 is planar when, equals ai or (3i, the eigenvaluessatisfy the equation

- det(A - ,1) = ,3 - ,2trA +, spmA - detA = o. (2.107)Because both right and left eigenvalues satisfy the same problem, they havethe same values which are denoted by~i. There are either one, two or threedifferent values for Cti. Because b i A aj = (aihi) aj = hi (ajaj), wesee that if Cti f; Ctj, the left and right eigenvectors are orthogonal, i.e. theysatisfy b, . aj = o.

Eigenvectors hi and ai corresponding to a solution Cti of (2.107) can beconstructed using dyadic methods. The construction depends on the mul-~tude~f the particular eigenvalue, which depends on whether the dyadicA - ail is strictly planar, strictly linear or null. Let us consider these casesseparately. For this we need the following identities:

a x (A~B) = B x (a A) +A x (a B),(A~B) x a = (:4.a) x B + (B a) x :4,

(2.108)

(2.109)

2.6. THE EIGENVALUE PROBLEM

with the special cases

a x (A~A) = 2A x (a A),(A~A) x a = 2(A a) x A.

37

(2.110)

(2.111)These can be derived with the general method described in Section 2.3, forcreating dyadic identities.

Strictly planar A - aJ Defining B, =JA - aJ)~(A - aJ) i= 0,from (2.110), (2.111) we see that hi x B; = B i x a, = 0, whencethere exists a scalar ei i= 0~uch that B; = eibiai. Thus, from theknowledge of Oi the dyadic B i is known and the eigenvectors can bewritten in the form a, = C B, and hi = B i . c with suitable vectorc. In this case Oi is a single root of (2.107).

Strictly linear -'!- oil. There exist non-null vectors c, d such that Ais of the form 0.1+cd. This is a special type of dyadic, which is calleduniaxial. (2.107) leaves us with the equation (;'-0)2(,-0-cd) = 0,which shows us that the eigenvalue Qi = Q is a double root of (2.107).Assuming c . d -# 0, the third root is a simple one, Qj = Q + c . d,for which the eigenvectors can be obtained through the expressionabove. The left and right eigenvectors corresponding to the doubleroot are any vectors satisfying the conditions b . c = 0 and d . a =0, respectively. For c . d = 0, all three eigenvalues are the same,but there only exist two linearly independent eigenvectors, those justmentioned.

Null dyadic A - Oil. In this case A = aI, there is a triple eigenvalue and any vector i~ an eigenvector. This happens if A is a multipleof the unit dyadic 1.

The previous classification was made in terms of the dyadic A - ail, ormultitude of a particular eigenvalue ai. Let us now consider the numberof different eigenvalues of a dyadic iI, which can b~ 1, 2 or 3. BecausedetA = a10203, spmA = {}102+02Q3+a301 and trA = a1 +a2+a3, theCayley-Hamilton equation (2.65) can be written as

(2.112)

The order of terms is immaterial here.

38 CHAPTER 2. DYAD/CS

One eigenvalue 01 = 02 == 03. Because (A -=-01/)3_= 0, from thesolution of equation (2.100) we conclude th~t 11 - ~II is a trace-freeshearer, whence the most general form for A is 01I.+ (ab + cc) x a.Taking the trace operation it is seen that 01 = trA/3. The numberof eigenvectors is obviously that of the shearer term. Here we canseparate the cases.

- Strictly planar trace-free shearer with a . b x c :f:. o. It is easyto show that there exists only one eigenvector i!eft a~ ri~t),which can be obtained from the expression (11 - 011) ~ (A -oIl) = 2(a . b x c)(a x c)a. Thus, the left eigenvector is a x cand the right eigenvector a. Only this type of dyadic has justone eigenvector.

- Linear shearer, w..Eich c~n be written with c = 0 in the aboveexpression. Now 11 - all satisfies (2.99). There exist two eigen-vectors, which are orthogonal to vectors a from the left and b x afrom the right.

Two eigenvalues a1 =F a2 = a3. The dyadic A satisfies an equationof the form (2.103): B2(B - trB I) = 0 with B = A- 02/, as is~asily verified. Thus..!. the dyadic B must be a general shearer andA is of the form 021 + a x (be + da). The eigenvalue 01 equalsa2 + a x b . c, whence the shearer here cannot be trace free in orderthat the two eigenvalues do not coincide. In this case there exist twolinearly independent eigenvectors.

Three eigenvalues 01 :f:. 02 :f:. a3. In this case there exist three linearlyindependent eigenvectors. In fact, assuming the eigenvector a3 to bea linear combination of a1 and a2, which are linearly independent,(2.105) will lead to the contradictory conditions a3 =aI, a3 = a2.

As a summary, the following table presents the different cases of dyadicswith different numbers of eigenvalues (N in horizontal lines) and eigenvec-tors (M in vertical columns).

MIN 1 2 31 01 + (ab + cc) x a none none2 0/ + aa x b 0/ + (ab + cd) x a none3 0/ 0/ +ab ab + cd + ef

2.7. HERMITIAN AND POSITIVE DEFINITE DYADICS 39

Because the left or right eigenvectors of a dyadic with three linearlyindependent eigenvectors form two bases {hi}, {ail, the dyadic can bewritten in either base as

(2.113)

From this we conclude that the left and right eigenvectors are in fact re-ciprocals of each other: ai = hi, hi = ai. If two dyadics commute, theyhave the same eigenvectors.

2.7 Hermitian and positive definite dyadics

Hermitian and positive definite dyadics are often encountered in electro-magnetics. In fact, lossless medium parameter dyadics are hermitian orantihermitian depending on the definition. Also, from power considera-tions in a medium, positive definiteness of dyadics often follows.

2.7.1 Hermitian dyadics

By definition, the hermitian dyadic satisfies

fIT=A*,whereas the antihermitian dyadic is defined by

AT = -A*.

(2.114)

(2.115)

Any dyadic can be written as a sum of a hermitian and an antihermitiandyadic in the form

(2.116)

Any hermitian dyadic can be written in the form E cc* and antihermi-tian, in the form L: jcc*. Conversely, these kinds of dyadics are alwayshermitian and antihermitian, respectively.

From (2.114), (2.115) it follows that the symmetric part of a hermitiandyadic is real and the antisymmetric part imaginary, whence the mostgeneral hermitian dyadic H can be written in the form

(2.117)

with real and symmetric S and real h. Any antihermitian dyadic can bewritten as jH, where H is a hermitian dyadic.


Hermitian dyadics form a subspace in the linear space of dyadics. Thedot product of two hermitian dyadics is not necessarily hermitian, but thedouble-cross product is, as is seen from

(2.118)

where A and B are hermitian. Also, the double-dot product of two hermi-tian dyadics is a re~ num~r, asJ.s easy to s~ from the sum expression.Thus, the scalars trA, spmA, detA are real if A is hermitian. This impliesthat the inverse of a hermitian dyadic is hermitian.

The following theorem is very useful when deriving identities for her-mitian dyadics:

= * =A : aa = 0 for all a, =} A = o. (2.119)

This can be pr~ed by setting first a = b + e and then a = b + je, whencethe condition if : be = 0 for all ~ectors b, c will result fro~ (2.119),making the matr~ components of if vanish. For comparison, A : aa = 0for all a implies A antisymmetric, as is easy to prove. From (2.119) wecan show that if A; aa" is real for all vectors a, A is hermitian. In fact,this implies A: aa" - A* : a*a = 0 or (A - A*T) : aa" = 0, whence A ishermitian.

A hermitian dyadic always has three eigenvectors no matter how manyeigenvalues it has, as can be shown. The right and left eigenvectors cor-responding to the same eigenvalues are complex conjugates of each other,because from A a == oa we have A* .a" = a" .it == a*a*. But eigenvaluesare real and eigenvectors conjugate orthogonal, because (ai -aj)ai .aj = 0,whence ai -a1 = 0 and a, .a; = 0 for ai =f. aj. Thus, the general hermitiandyadic can be written in terms of its eigenvalues and eigenvectors as

(2.120)

2.7.2 Positive definite dyadics

By definition, a dyadic D is positive definite (PD), if it satisfies

D : aa" > 0, for all a f:. o. (2.121 )

A PD dyadic is always hermitian, as is evident. Other properties, whoseproofs are partly omitted, follow.

2.8. SPECIAL DYADICS 41

PD dyadics are complete. If D were planar, there would exist a vectora such that Ir-e. = 0, in.contradiction with (2.121). Thus, the inverseof a PD dyadic always exists.

PD dyadics possess positive eigenvalues. This is seen by dot multi-plying the expansion (2.120) by ajaj/aj . aj, and the result is aj,which must be greater than 0 because of (2.121).

if A is PD, its symmetric part is PD.

A is PD exactly when its invariants trA, spmA, detA are real andpositive.

Aand B PD implies A~B PD. A dyadic of the form A.A*T is positive semidefinite and PD if A is

complete.

2.8 Special dyadics

In this section we consider some special classes of dyadics appearing in prac-tical electromagnetic problems. Rotation and reflection dyadics emerge insymmetries of various structures whereas uniaxial and gyrotropic dyadicsare encountered when electromagnetic fields in special materials are anal-ysed. Parameters of some media like the sea ice can be approximated interms of a uniaxial dyadic, while others like magnetized ferrite or magne-toplasma may exhibit properties which can be analysed using gyrotropicdyadics.

2.8.1 Rotation dyadics

In real vector space, the rotation of a vector by an angle () in the right-handdirection around the axis defined by the unit vector u can be written interms of the following dyadic:

R(u, 0) = uu + sin O(u x I) + cos (}(I - uu), (2.122)

It can also be written in the form eu xIB, as is seen if the dyadic exponen-tial function is written as a Taylor series. The rotation dyadic obeys theproperties

R(-u,O) = R(u, -B) = RT(u, 0) = :R-1 (u , 0), (2.123)


R(U, ( 1) . R(u, ( 2 ) == R(u, 81 + (2 ) ,R(U, 8)~R(u, (J) == 2R(u, 8),

- 1- -detR(u, (J) == "3 R(u, 8) : R(u, (J) == 1.

(2.124)(2.125)

(2.126)

It is not difficult to show that the propertie~detR == 1 and RT == R,-luniquely define the form (2.122) of the dyad~ R(u, (J), so that they couldbe given as the definition. Also, (2.125) with R 1= 0 would do. In general, (Jand/or u may be complex, which means that the geometrical interpretationis lost. The resulting dyadic is also called the rotation dyadic in the complexcase.

The dot product of two rotation dyadics with arbitrary axes and anglesis another rotation dyadic. This is seen from

- - - - 1- - - - --(R1 R2)~(Rl . R2) = 2(Rl~Rd (R2~R2) = 2(R1 R2), (2.127)where use has been made of the identity (2.67). It is not very easy to findthe axis and angle of the resulting rotation dyadic. This can be done per-haps most easily by using a representation in terms of a special gyrotropicdyadic. The gyro tropic dyadic was defined in (2.35) and it is the most gen-eral non-symmetric dyadic which commutes with its own transpose. Thespecial gyrotropic dyadic of interest here is of the type

- -

G(q) = I + q x I,and the rotation dyadic can be written as

In fact, because

(2.128)

(2.129)

(2.130)- - 1 - -(1- q x 1)-1 = --2 (1+ qq + q x 1),l+q

(2.129) can be seen to be of the form (2.122) if we write q == u tan(B/2),or u == q/.;q:q and (J = 2 tan-l.;q:q. Thus, q is not a CP vector. Forreal q, its length q determines the angle of rotation. q = 0 corresponds toB == 0, q == 00 to (J == 1r and q == 1 to () == 1r/2.

It is straightforward, although a bit tedious, to prov~ th!. followingidentity between the dot product of two rotation dyadics HI, R2 and thecorresponding q vectors:

(R R) ql + q2 + q2 X qlq 1 2 == .1 - qt . q2

(2.131 )

2.8. SPECIAL DYADICS 43

This~quation shows us that two rotations do not commute in general, sinceR 1 . R 2 and R2 . R 1 lead to the same q vector only if q1 and q2 are parallelvectors, i.e. the two rotation dyadics have the same axes.

2.8.2 Reflection dyadics

The symmetric dyadic of the form

C{u) = I - 2uu (2.132)with a unit vector u is called the reflectio~dyadic because, when u is real,mapping the position vector r through C . r = r - 2u(u . r) obviouslyperforms a reflection in the plane u . r == O. The reflection dyadic can bealso presented as a negative rotation by an angle 1r around u as the axis

C(u) = -R(u,1r), (2.133)as is seen from the definition of the rotation dyadic (2.122). In fact, theunit dyadic 1 and the negative of the reflection dyadic are the only rotationdyadics that are symmetric.

The reflection dyadic satisfies

C 2(u) = I, or C -1 = C, (2.134)C(u)~C(u) == -2C(u), (2.135)

- - -

trC = 1, spmC = -1, detC = -1, (2.136)The most general square roo!. of the unit dyadic is not the reflection dyadic,but a dyadic of the form (1 - 2ab) with either a b = 1 or ab = o.

It is easy to see that both rotation and reflection dyadics preserve theinner product of two vectors. In fact, because they both satisfy AT .A == I,we have

(A . a) . (A. b) = a- (AT. A) .b = a b, (2.137)for any vectors a, b. The cross product is transformed differently throughrotation

- - 1-- -(R. a) x (R b) = -(R~R) . (a x b) == R (a x b),2

than through reflection

(C a) x (C b) = -C (a x b).

(2.138)

(2.139)This equation shows us that a reflection transformed electromagnetic field isnot an electromagnetic field, because if the electric and magnetic fields aretransformed through reflection, the Poynting vector is not. The converseis, however, true for the rotation transformation.


2.8.3 Uniaxial dyadics

By definition, a uniaxial dyadic is of the general form- -

D = aI + abo (2.140)

A condition for !} to be uniaxial is obviously that there exist a scalar asuch that D - al is a linear dyadic, or

- -(D - al)~(D - aI) = o. (2.141)It is easi~ to define conditions not for D itself but for its trace-free part

C = r:- trpi. D is uniaxial exactly when C is uniaxial, or of the formC = {31 + ab, with (J = -a . b/3, whence it satisfies

(C - (3I)~(C - (3I) = o.Taking the trace operation leaves us an equation for {3:

= = -T(32 _ spmC _ C : C---3---6-

(2.142)

(2.143)

(2.144)

Either root of (2.143) inserted in (2.142) would result in a dyadic equationwhich is satisfied for any trace-free uniaxial dyadic C. Applying (2.50) and(2.51) gi

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