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Geometry and Topology
Geometry provides a whole range of views on the universe, serving as the inspiration, technical
toolkit and ultimate goal for many branches of mathematics and physics. This book introduces
the ideas of geometry, and includes a generous supply of simple explanations and examples.
The treatment emphasises coordinate systems and the coordinate changes that generate symme-
tries. The discussion moves from Euclidean to non-Euclidean geometries, including spherical
and hyperbolic geometry, and then on to af¬ne and projective linear geometries. Group theory
is introduced to treat geometric symmetries, leading to the uni¬cation of geometry and group
theory in the Erlangen program. An introduction to basic topology follows, with the M¨ biuso
strip, the Klein bottle and the surface with g handles exemplifying quotient topologies and
the homeomorphism problem. Topology combines with group theory to yield the geometry
of transformation groups, having applications to relativity theory and quantum mechanics. A
¬nal chapter features historical discussions and indications for further reading. While the book
requires minimal prerequisites, it provides a ¬rst glimpse of many research topics in modern
algebra, geometry and theoretical physics.
The book is based on many years™ teaching experience, and is thoroughly class tested.
There are copious illustrations, and each chapter ends with a wide supply of exercises. Further
teaching material is available for teachers via the web, including assignable problem sheets
with solutions.

m i l e s r e i d is a Professor of Mathematics at the Mathematics Institute, University of Warwick
b a l a zs szendro is a Faculty Lecturer in the Mathematical Institute, University of Oxford,
´ ´´i
and Martin Powell Fellow in Pure Mathematics at St Peter™s College, Oxford
Geometry and
Topology

Miles Reid
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Bal´zs Szendro
a ´´i
Mathematical Institute, University of Oxford,
24“29 St Giles, Oxford OX1 3LB, UK
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521848893

© Cambridge University Press 2005


This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.

First published in print format 2005

isbn-13 978-0-511-13733-4 eBook (NetLibrary)
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Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents




List of ¬gures page x
Preface xiii

1 Euclidean geometry 1
1.1 The metric on Rn 1
1.2 Lines and collinearity in Rn 3
1.3 Euclidean space En 4
1.4 Digression: shortest distance 4
1.5 Angles 5
1.6 Motions 6
1.7 Motions and collinearity 7
1.8 A motion is af¬ne linear on lines 7
1.9 Motions are af¬ne transformations 8
1.10 Euclidean motions and orthogonal transformations 9
1.11 Normal form of an orthogonal matrix 10
1.11.1 The 2 — 2 rotation and re¬‚ection matrixes 10
1.11.2 The general case 12
1.12 Euclidean frames and motions 14
1.13 Frames and motions of E2 14
1.14 Every motion of E2 is a translation, rotation, re¬‚ection or glide 15
1.15 Classi¬cation of motions of E3 17
1.16 Sample theorems of Euclidean geometry 19
1.16.1 Pons asinorum 19
1.16.2 The angle sum of triangles 19
1.16.3 Parallel lines and similar triangles 20
1.16.4 Four centres of a triangle 21
1.16.5 The Feuerbach 9-point circle 23
Exercises 24
2 Composing maps 26
2.1 Composition is the basic operation 26
2.2 Composition of af¬ne linear maps x ’ Ax + b 27

v
vi CONTENTS


2.3 Composition of two re¬‚ections of E2 27
2.4 Composition of maps is associative 28
2.5 Decomposing motions 28
2.6 Re¬‚ections generate all motions 29
2.7 An alternative proof of Theorem 1.14 31
2.8 Preview of transformation groups 31
Exercises 32
3 Spherical and hyperbolic non-Euclidean geometry 34
3.1 Basic de¬nitions of spherical geometry 35
3.2 Spherical triangles and trig 37
3.3 The spherical triangle inequality 38
3.4 Spherical motions 38
3.5 Properties of S 2 like E2 39
3.6 Properties of S 2 unlike E2 40
3.7 Preview of hyperbolic geometry 41
3.8 Hyperbolic space 42
3.9 Hyperbolic distance 43
3.10 Hyperbolic triangles and trig 44
3.11 Hyperbolic motions 46
3.12 Incidence of two lines in H2 47
3.13 The hyperbolic plane is non-Euclidean 49
3.14 Angular defect 51
3.14.1 The ¬rst proof 51
3.14.2 An explicit integral 51
3.14.3 Proof by subdivision 53
3.14.4 An alternative sketch proof 54
Exercises 56
4 Affine geometry 62
4.1 Motivation for af¬ne space 62
4.2 Basic properties of af¬ne space 63
4.3 The geometry of af¬ne linear subspaces 65
4.4 Dimension of intersection 67
4.5 Af¬ne transformations 68
4.6 Af¬ne frames and af¬ne transformations 68
4.7 The centroid 69
Exercises 69
5 Projective geometry 72
5.1 Motivation for projective geometry 72
5.1.1 Inhomogeneous to homogeneous 72
5.1.2 Perspective 73
5.1.3 Asymptotes 73
5.1.4 Compacti¬cation 75
CONTENTS vii


5.2 De¬nition of projective space 75
5.3 Projective linear subspaces 76
5.4 Dimension of intersection 77
5.5 Projective linear transformations and projective frames of reference 77
5.6 Projective linear maps of P1 and the cross-ratio 79
5.7 Perspectivities 81
5.8 Af¬ne space An as a subset of projective space Pn 81
5.9 Desargues™ theorem 82
5.10 Pappus™ theorem 84
5.11 Principle of duality 85
5.12 Axiomatic projective geometry 86
Exercises 88

6 Geometry and group theory 92
6.1 Transformations form a group 93
6.2 Transformation groups 94
6.3 Klein™s Erlangen program 95
6.4 Conjugacy in transformation groups 96
6.5 Applications of conjugacy 98
6.5.1 Normal forms 98
6.5.2 Finding generators 100
6.5.3 The algebraic structure of transformation groups 101
6.6 Discrete re¬‚ection groups 103
Exercises 104

7 Topology 107
7.1 De¬nition of a topological space 108
7.2 Motivation from metric spaces 108
7.3 Continuous maps and homeomorphisms 111
7.3.1 De¬nition of a continuous map 111
7.3.2 De¬nition of a homeomorphism 111
7.3.3 Homeomorphisms and the Erlangen program 112
7.3.4 The homeomorphism problem 113
7.4 Topological properties 113
7.4.1 Connected space 113
7.4.2 Compact space 115
7.4.3 Continuous image of a compact space is compact 116
7.4.4 An application of topological properties 117
7.5 Subspace and quotient topology 117
7.6 Standard examples of glueing 118
7.7 Topology of Pn 121
R
7.8 Nonmetric quotient topologies 122
7.9 Basis for a topology 124
viii CONTENTS


7.10 Product topology 126
7.11 The Hausdorff property 127
7.12 Compact versus closed 128
7.13 Closed maps 129
7.14 A criterion for homeomorphism 130
7.15 Loops and the winding number 130
7.15.1 Paths, loops and families 131
7.15.2 The winding number 133
7.15.3 Winding number is constant in a family 135
7.15.4 Applications of the winding number 136
Exercises 137

8 Quaternions, rotations and the geometry of
transformation groups 142
8.1 Topology on groups 143
8.2 Dimension counting 144
8.3 Compact and noncompact groups 146
8.4 Components 148
8.5 Quaternions, rotations and the geometry of SO(n) 149
8.5.1 Quaternions 149
8.5.2 Quaternions and rotations 151
8.5.3 Spheres and special orthogonal groups 152
8.6 The group SU(2) 153
8.7 The electron spin in quantum mechanics 154
8.7.1 The story of the electron spin 154
8.7.2 Measuring spin: the Stern“Gerlach device 155
8.7.3 The spin operator 156
8.7.4 Rotate the device 157
8.7.5 The solution 158
8.8 Preview of Lie groups 159
Exercises 161

9 Concluding remarks 164
9.1 On the history of geometry 165
9.1.1 Greek geometry and rigour 165
9.1.2 The parallel postulate 165
9.1.3 Coordinates versus axioms 168
9.2 Group theory 169
9.2.1 Abstract groups versus transformation groups 169
9.2.2 Homogeneous and principal homogeneous spaces 169
9.2.3 The Erlangen program revisited 170
9.2.4 Af¬ne space as a torsor 171
CONTENTS ix


9.3 Geometry in physics 172
9.3.1 The Galilean group and Newtonian dynamics 172
9.3.2 The Poincar´ group and special relativity
e 173
9.3.3 Wigner™s classi¬cation: elementary particles 175
9.3.4 The Standard Model and beyond 176
9.3.5 Other connections 176
9.4 The famous trichotomy 177
9.4.1 The curvature trichotomy in geometry 177
9.4.2 On the shape and fate of the universe 178
9.4.3 The snack bar at the end of the universe 179

Appendix A Metrics 180
Exercises 181

Appendix B Linear algebra 183
B.1 Bilinear form and quadratic form 183
B.2 Euclid and Lorentz 184
B.3 Complements and bases 185
B.4 Symmetries 186
B.5 Orthogonal and Lorentz matrixes 187
B.6 Hermitian forms and unitary matrixes 188
Exercises 189

References 190
Index 193
Figures




A coordinate model of space page xiv
1.1 Triangle inequality 2
1.5 Angle with direction 6
1.6 Rigid body motion 6
Af¬ne linear construction of »x + µy
1.9 9
1.11a A rotation in coordinates 11
1.11b The rotation and the re¬‚ection 11
The Euclidean frames P0 , P1 , P2 and P0 , P1 , P2
1.13 14
Rot(O, θ ) and Glide(L, v)
1.14a 15
1.14b Construction of glide 15
1.14c Construction of rotation 16
Twist (L, θ, v) and Rot-Re¬‚ (L, θ, )
1.15a 17
1.15b A grid of parallel planes and their orthogonal lines 17
1.16a Pons asinorum 19
Sum of angles in a triangle is equal to π
1.16b 20
1.16c Parallel lines fall on lines in the same ratio 20
1.16d Similar triangles 21
1.16e The centroid 21
1.16f The circumcentre 22
1.16g The orthocentre 22
1.16h The Feuerbach 9-point circle 23

2.3 Composite of two re¬‚ections 28
2.7 Composite of a rotation and a re¬‚ection 31

3.0 Plane-like geometry 35
3.2 Spherical trig 38
Overlapping segments of S 2 41
3.6
The hyperbola t 2 = 1 + x 2 and t > 0
3.7 42
Hyperbolic space H2
3.8 43


x
LIST OF FIGURES xi


3.10 Hyperbolic trig 45
3.12 (a) Projection to the (x, y)-plane of the spherical lines y = cz
(b) Projection to the (x, y)-plane of the hyperbolic lines y = ct 48
3.13 The failure of the parallel postulate in H2 50
3.14a The hyperbolic triangle PQR with one ideal vertex 52
3.14b Area and angle sums are ˜additive™ 52
3.14c The subdivision of PQR. 54
3.14d The angular defect formula 55
3.14e Area is an additive function 56
3.14f Area is a monotonic function 56
3.15 H-lines 60

4.2 Points, vectors and addition 64
4.3a The af¬ne construction of the line segment [p, q] 66
4.3b Parallel hyperplanes 66
4.7 The af¬ne centroid 70
4.8 A weighted centroid 70

5.1a A cube in perspective 74
5.1b Perspective drawing 74
5.1c Hyperbola and parabola 74
The 3-transitive action of PGL(2) on P1
5.6a 80
The cross-ratio {P, Q; R, S}
5.6b 80
The inclusion An ‚ Pn
5.8 82
The Desargues con¬guration in P2 or P3
5.9a 83
Lifting the Desargues con¬guration to P3
5.9b 84
5.10 The Pappus con¬guration 85
5.12a Axiomatic projective plane 87
5.12b Geometric construction of addition 88

6.0 The plan of Coventry market 93
The conjugate rotation g Rot(P θ )g ’1 = Rot(g(P), g(θ))
,
6.4a 97
Action of Aff(n) on vectors of An
6.4b 98
6.6a Kaleidoscope 104
6.6b ˜Mus´ e Gr´ vin™
e e 104

7.2a Hausdorff property 110
S 1 = [0, 1] with the ends identi¬ed
7.2b 110
(0, 1) R
7.3a 112
7.3b Squaring the circle 112
7.4a Path connected set 114
7.6a The M¨ bius strip M
o 119
The cylinder S 1 — [0, 1]
7.6b 119
xii LIST OF FIGURES


7.6c The torus 120
7.6d Surface with g handles 120
7.6e Boundary and interior points 121
Topology of P2 : M¨ bius strip with a disc glued in
7.7 o 122
R
7.8a The mousetrap topology 123
Equivalence classes of quadratic forms ax 2 + 2bx y + cy 2
7.8b 124
7.10 Balls for product metrics 126
7.12 Separating a point from a compact subset 128
7.13a Closed map 129
7.13b Nonclosed map 129
7.15a Continuous family of paths 131
D— covered by overlapping open radial sectors
7.15b 134
7.15c Overlapping intervals 134
7.16a Glueing patterns on the square 140
7.16b The surface with two handles and the 12-gon 140

8.0 The geometry of the group of planar rotations 143
8.7a The Stern“Gerlach experiment 155
8.7b The modi¬ed Stern“Gerlach device 156
8.7c Two identical SG devices 156
8.7d Two different SG devices 157

9.1a The parallel postulate. To meet or not to meet? 166
9.1b The parallel postulate in the Euclidean plane 166
9.1c The ˜parallel postulate™ in spherical geometry 168
9.4a The cap, ¬‚at plane and Pringle™s chip 178
The genus trichotomy g = 0, g = 1, g ≥ 2 for oriented surfaces
9.4b 178

A.1 The bear 182
Preface




What is geometry about?
Geometry ˜measuring the world™ attempts to describe and understand space around
us and all that is in it. It is the central activity and main driving force in many branches
of math and physics, and offers a whole range of views on the nature and meaning
of the universe. This book treats geometry in a wide context, including a wealth of
relations with surrounding areas of math and other aspects of human experience.
Any discussion of geometry involves tension between the twin ideals of intuition
and precision. Descriptive or synthetic geometry takes as its starting point our ideas
and experience of the observed world, and treats geometric objects such as lines and
shapes as objects in their own right. For example, a line could be the path of a light
ray in space; you can envisage comparing line segments or angles by ˜moving™ one
over another, thus giving rise to notions of ˜congruent™ ¬gures, equal lengths, or equal
angles that are independent of any quantitative measurement. If A, B, C are points
along a line segment, what it means for B to be between A and C is an idea hard-wired
into our consciousness. While descriptive geometry is intuitive and natural, and can
be made mathematically rigorous (and, of course, Euclidean geometry was studied in
these terms for more than two millennia, compare 9.1), this is not my main approach
in this book.
My treatment centres rather on coordinate geometry. This uses Descartes™ idea
(1637) of measuring distances to view points of space and geometric quantities in
terms of numbers, with respect to a ¬xed origin, using intuitive ideas such as ˜a bit
to the right™ or ˜a long way up™ and using them quantitatively in a systematic and
precise way. In other words, I set up the (x, y)-plane R2 , the (x, y, z)-space R3 or
whatever I need, and use it as a mathematical model of the plane (space, etc.), for
the purposes of calculations. For example, to plan the layout of a car park, I might
map it onto a sheet of paper or a computer screen, pretending that pairs (x, y) of real
numbers correspond to points of the surface of the earth, at least in the limited region
for which I have planning permission. Geometric constructions, such as drawing an
even rectangular grid or planning the position of the ticket machines to ensure the
maximum aggravation to customers, are easier to make in the model than in real


xiii
xiv PREFACE




z

x
y


A coordinate model of space.


life. We admit possible drawbacks of our model, but its use divides any problem into
calculations within the model, and considerations of how well it re¬‚ects the practical
world.
Topology is the youngster of the geometry family. Compared to its venerable
predecessors, it really only got going in the twentieth century. It dispenses with
practically all the familiar quantities central to other branches of geometry, such
as distance, angles, cross-ratios, and so on. If you are tempted to the conclusion
that there is not much left for topology to study, think again. Whether two loops of
string are linked or not does not depend on length or shape or perspective; if that
seems too simple to be a serious object of study, what about the linking or knotting
of strands of DNA, or planning the over- and undercrossings on a microchip? The
higher dimensional analogues of disconnecting or knotting are highly nontrivial and
not at all intuitive to denizens of the lower dimensions such as ourselves, and cannot
be discussed without formal apparatus. My treatment of topology runs brie¬‚y through
abstract point-set topology, a fairly harmless generalisation of the notion of continuity
from a ¬rst course on analysis and metric spaces. However, my main interest is in
topology as rubber-sheet geometry, dealing with manifestly geometric ideas such as
closed curves, spheres, the torus, the M¨ bius strip and the Klein bottle.
o


Change of coordinates, motions, group theory
and the Erlangen program
Descartes™ idea to use numbers to describe points in space involves the choice of
a coordinate system or coordinate frame: an origin, together with axes and units of
length along the axes. A recurring theme of all the different geometries in this book
is the question of what a coordinate frame is, and what I can get out of it. While
coordinates provide a convenient framework to discuss points, lines, and so on, it
is a basic requirement that any meaningful statement in geometry is independent of
the choice of coordinates. That is, coordinate frames are a humble technical aid in
determining the truth, and are not allowed the dignity of having their own meaning.
Changing from one coordinate frame to another can be viewed as a transformation
or motion: I can use a motion of space to align the origin and coordinate axes of two
coordinate systems. A statement that remains true under any such motion is indepen-
dent of the choice of coordinates. Felix Klein™s 1872 Erlangen program formalises
PREFACE xv


this relation between geometric properties and changes of coordinates by de¬ning
geometry to be the study of properties invariant under allowed coordinate changes,
that is, invariant under a group of transformations. This approach is closely related to
the point of view of special relativity in theoretical physics (Einstein, 1905), which
insists that the laws of physics must be invariant under Lorentz transformations.
This course discusses several different geometries: in some case the spaces them-
selves are different (for example, the sphere and the plane), but in others the differ-
ence is purely in the conventions I make about coordinate changes. Metric geometries
such as Euclidean and hyperbolic non-Euclidean geometry include the notions of dis-
tance between two points and angle between two lines. The allowed transformations
are rigid motions (isometries or congruences) of Euclidean or hyperbolic space. Af¬ne
and projective geometries consider properties such as collinearity of points, and the
typical group is the general linear group GL(n), the group of invertible n — n ma-
trixes. Projective geometry presents an interesting paradox: while its mathematical
treatment involves what may seem to be quite arcane calculations, your brain has a
sight driver that carries out projective transformations by the thousand every time
you recognise an object in perspective, and does so unconsciously and practically
instantaneously.
The sets of transformations that appear in topology, for example the set of all
continuous one-to-one maps of the interval [0, 1] to itself, or the same thing for the
circle S 1 or the sphere S 2 , are of course too big for us to study by analogy with trans-
formation groups such as GL(n) or the Euclidean group, whose elements depend on
¬nitely many parameters. In the spirit of the Erlangen program, properties of spaces
that remain invariant under such a huge set of equivalences must be correspondingly
coarse. I treat a few basic topological properties such as compactness, connectedness,
winding number and simple connectedness that appear in many different areas of
analysis and geometry. I use these simple ideas to motivate the central problem of
topology: how to distinguish between topologically different spaces? At a more ad-
vanced level, topology has developed systematic invariants that apply to this problem,
notably the fundamental group and homology groups. These are invariants of spaces
that are the same for topologically equivalent spaces. Thus if you can calculate one
of these invariants for two spaces (for example, a disc and a punctured disc) and
prove that the answers are different, then the spaces are certainly not topologically
equivalent. You may want to take subsequent courses in topology to become a real
expert, and this course should serve as a useful guide in this.


Geometry in applications
Although this book is primarily intended for use in a math course, and the topics are
oriented towards the theoretical foundations of geometry, I must stress that the math
ideas discussed here are applicable in different ways, basic or sophisticated, as stated
or with extra development, on their own or in combination with other disciplines,
Euclidean or non-Euclidean, metric or topological, to a huge variety of scienti¬c and
technological problems in the modern world. I discuss in Chapter 8 the quantum
xvi PREFACE


mechanical description of the electron that illustrates a fundamental application of
the ideas of group theory and topology to the physics of elementary particles. To
move away from basic to more applied science, let me mention a few examples
from technology. The typesetting and page layout software now used throughout the
newspaper and publishing industry, as well as in the computer rooms of most univer-
sity departments, can obviously not exist without a knowledge of basic coordinate
geometry: even a primary instruction such as ˜place letter A or box B, scaled by such-
and-such a factor, slanted at such-and-such an angle, at such-and-such a point on the
page™ involves af¬ne transformations. Within the same industry, computer typefaces
themselves are designed using Bezier curves. The geometry used in robotics is more
sophisticated. The technological aim is, say, to get a robot arm holding a spanner into
the right position and orientation, by adjusting some parameters, say, angles at joints
or lengths of rods. This translates in a fairly obvious way into the geometric prob-
lem of parametrising a piece of the Euclidean group; but the solution or approximate
solution of this problem is hard, involving the topology and analysis of manifolds,
algebraic geometry and singularity theory. The computer processing of camera im-
ages, whose applications include missile guidance systems, depends among other
things on projective transformations (I say this for the bene¬t of students looking
for a career truly worthy of their talents and education). Although scarcely having
the same nobility of purpose, similar techniques apply in ultrasonic scanning used
in antenatal clinics; here the geometric problem is to map the variations in density
in a 3-dimensional medium onto a 2-dimensional computer screen using ultrasonic
radar, from which the human eye can easily make out salient features. By a curious
coincidence, 3 hours before I, the senior author, gave the ¬rst lecture of this course in
January 1989, I was at the maternity clinic of Walsgrave hospital Coventry looking
at just such an image of a 16-week old foetus, now my third daughter Murasaki.


About this book
This book is intended for the early years of study of an undergraduate math course.
Who the
For the most part, it is based on a second year module taught at Warwick over many
book is for
years, a module that is also taken by ¬rst and third year math students, and by students
from the math/physics course. You will ¬nd the book accessible if you are familiar
with most of the following, which is standard material in ¬rst and second year math
courses.


How to express lines and circles in R2 in terms of coordi-
Coordinate geometry
nates, and calculate their points of intersection; some idea of how to do the same in
R3 and maybe Rn may also be helpful.


Vector spaces and linear maps over R and C, bases and matrixes,
Linear algebra
change of bases, eigenvalues and eigenvectors. This is the only major piece of math
that I take for granted. The examples and exercises make occasional reference to
PREFACE xvii


vector spaces over ¬elds other than R or C (such as ¬nite ¬elds), but you can always
omit these bits if they make you uncomfortable.

Bilinear and quadratic forms, and how to express them in ma-
Multilinear algebra
trix terms; also Hermitian forms. I summarise all the necessary background material
in Appendix B.

Some prior familiarity with the ¬rst ideas of a metric space course
Metric spaces
would not do any harm, but this is elementary material, and Appendix A contains all
that you need to know.

I have gone to some trouble to develop from ¬rst principles all
Group theory
the group theory that I need, with the intention that my book can serve as a ¬rst
introduction to transformation groups and the notions of abstract group theory if you
have never seen these. However, if you already have some idea of basic things such
as composition laws, subgroups, cosets and the symmetric group, these will come in
handy as motivation. If you prefer to see a conventional introduction to group theory,
there are any number of textbooks, for example Green [10] or Ledermann [14]. If you
intend to study group theory beyond the introductory stage, I strongly recommend
Artin [1] or Segal [22]. My ideological slant on this issue is discussed in more detail
in 9.2.

Although the thousands queueing impatiently at supermarkets and airport bookshops
How to use
to get their hands on a copy of this book for vacation reading was strong motivation
the book
for me in writing it, experience suggests the harsher view of reality: at least some of
my readers may bene¬t from coercion in the form of an organised lecture course.
Experience from teaching at Warwick shows that Chapters 1“6 make a reasonably
paced 30 hour second year lecture course. Some more meat could be added to subjects
that the lecturer or students ¬nd interesting; re¬‚ection groups following Coxeter [5],
Chapter 4 would be one good candidate. Topics from Chapters 7“8 or the further
topics of Chapter 9 could then pro¬tably be assigned to students as essay or project
material. An alternative course oriented towards group theory could start with af¬ne
and Euclidean geometry and some elements of topology (maybe as a refresher), and
concentrate on Chapters 3, 6 and 8, possibly concluding with some material from
Segal [22]. This would provide motivation and techniques to study matrix groups
from a geometric point of view, one often ignored in current texts.

I want the book to be as informal as possible in style. To this end, I always refer
The author™s
to the student as ˜you™, which has the additional advantage that it is independent of
identity
your gender and number. I also refer to myself by the ¬rst person singular, despite
crisis
the fact that there are two of me. Each of me has lectured the material many times,
and is used to taking personal responsibility for the truth of my assertions. My model
is van der Waerden™s style, who always wrote the crisp ˜Ich behaupte . . . ™ (often
when describing results he learned from Emmy Noether or Emil Artin™s lectures). I
xviii PREFACE


leave you to imagine the speaker as your ideal teacher, be it a bearded patriarch or a
fresh-faced bespectacled Central European intellectual.

Acknowledge- A second year course with the title ˜Geometry™ or ˜Geometry and topology™ has
been given at Warwick since the 1960s. It goes without saying that my choice of
ments
material, and sometimes the material itself, is taken in part from the experience of
colleagues, including John Jones, Colin Rourke, Brian Sanderson; David Epstein has
also provided some valuable material, notably in the chapter on hyperbolic geom-
etry. I have also copied material consciously or unconsciously from several of the
textbooks recommended for the course, in particular Coxeter [5], Rees [19], Nikulin
and Shafarevich [18] and Feynman [7]. I owe special thanks to Katrin Wendland, the
most recent lecturer of the Warwick course MA243 Geometry, who has provided a
detailed criticism of my text, thereby saving me from a variety of embarrassments.

Wen solche Lehren nicht erfreun,
Disclaimer
Verdienet nicht ein Mensch zu sein.
From Sarastro™s aria, The Magic Flute, II.3.
This is an optional course. If you don™t like my teaching, please deregister before the
deadline.
1 Euclidean geometry




This chapter discusses the geometry of n-dimensional Euclidean space En , together
with its distance function. The distance gives rise to other notions such as angles and
congruent triangles. Choosing a Euclidean coordinate frame, consisting of an origin
O and an orthonormal basis of vectors out of O, leads to a description of En by
coordinates, that is, to an identi¬cation En = Rn .
A map of Euclidean space preserving Euclidean distance is called a motion or rigid
body motion. Motions are fun to study in their own right. My aims are

(1) to describe motions in terms of linear algebra and matrixes;
(2) to ¬nd out how many motions there are;
(3) to describe (or classify) each motion individually.

I do this rather completely for n = 2, 3 and some of it for all n. For example, the
answer to (2) is that all points of En , and all sets of orthonormal coordinate frames at
a point, are equivalent: given any two frames, there is a unique motion taking one to
the other. In other words, any point can serve as the origin, and any set of orthogonal
axes as the coordinate frames. This is the geometric and philosophical principle that
space is homogeneous and isotropic (the same viewed from every point and in every
direction). The answer to (3) in E2 is that there are four types of motions: translations
and rotations, re¬‚ections and glides (Theorem 1.14).
The chapter concludes with some elementary sample theorems of plane Euclidean
geometry.

The metric on Rn
1.1
Throughout the book, I write Rn for the vector space of n-tuples (x1 , . . . , xn ) of real
numbers. I start by discussing its metric geometry. The familiar Euclidean distance
function on Rn is de¬ned by
« «
x1 y1
¬.· ¬.·
|x ’ y| = (xi ’ yi )2 , where x =  .  and y =  . . (1)
. .
xn yn

1
2 EUCLIDEAN GEOMETRY


z



u v




x y
Figure 1.1 Triangle inequality.


The relationship between this distance function and the Euclidean inner product (or
dot product) x · y = xi yi on Rn is discussed in Appendix B.2. The more important
point is that the Euclidean distance (1) is a metric on Rn . If you have not yet met
the idea of a metric on a set X , see Appendix A; for now recall that it is a distance
function d(x, y) satisfying positivity, symmetry and the triangle inequality. Both the
positivity |x ’ y| ≥ 0 and symmetry |x ’ y| = |y ’ x| are immediate, so the point is
to prove the triangle inequality (Figure 1.1).

Theorem (Triangle inequality)

|x ’ y| ¤ |x ’ z| + |z ’ y|, for all x, y, z ∈ Rn , (2)

with equality if and only if z = x + »(y ’ x) for » a real number between 0 and 1.

Set x ’ z = u and z ’ y = v so that x ’ y = u + v; then (2) is equivalent
Proof
to

u i2 + v i2 ≥ (u i + v i )2 . (3)

Note that both sides are nonnegative, so that squaring, one sees that (3) is equivalent
to

u i2 + v i2 + 2 u i2 · v i2 ≥ (u i + v i )2

= u i2 + v i2 + 2 ui vi . (4)

Cancelling terms, one sees that (4) is equivalent to

u i2 · v i2 ≥ ui vi . (5)

If the right-hand side is negative then (5), hence also (2), is true and strict. If the
right-hand side of (5) is ≥ 0 then it is again permissible to square both sides, giving

u i2 · v2 ≥ u jv j .
ui vi (6)
j
1.2 LINES AND COLLINEARITY IN Rn 3


You will see at once what is going on if you write this out explicitly for n = 2 and
expand both sides. For general n, the trick is to use two different dummy indexes i, j
as in (6): expanding and cancelling gives that (6) is equivalent to

(u i v j ’ u j v i )2 ≥ 0. (7)
i> j

Now (7) is true, so retracing our steps back through the argument gives that (2) is
true. Finally, equality in (2) holds if and only if u i v j = u j v i for all i, j (from (7))
u i v i ≥ 0 (from the right-hand side of (5)); that is, u and v are proportional,
and
u = µv with µ ≥ 0. Rewriting this in terms of x, y, z gives the conclusion. QED


Lines and collinearity in Rn
1.2
There are several ways of de¬ning a line (already in the usual x, y plane R2 ); I choose
one de¬nition for Rn .


Let u ∈ Rn be a ¬xed point and v ∈ Rn a nonzero direction vector.
Definition
The line L starting at u ∈ Rn with direction vector v is the set

L := u + »v » ∈ R ‚ Rn .

Three distinct points x, y, z ∈ Rn are collinear if they are on a line.

If I choose the starting point x, and the direction vector v = y ’ x, then
L = {(1 ’ »)x + »y}. To say that distinct points x, y, z are collinear means that z =
{(1 ’ »)x + »y} for some ». Writing

[x, y] = x + »(y ’ x) 0 ¤ » ¤ 1

for the line segment between x and y, the possible orderings of x, y, z on the line L
are controlled by
 ±
  x ∈ [z, y]
» ¤ 0 
 
0 ¤ » ¤ 1 ⇐’ z ∈ [x, y]
 
 
1 ¤ »  y ∈ [x, z].

Together with the triangle inequality Theorem 1.1, this proves the following result.

Three distinct points x, y, z ∈ Rn are collinear if and only if (after a
Corollary
permutation of x, y, z if necessary)

|x ’ y| + |y ’ z| = |x ’ z|.

In other words, collinearity is determined by the metric.
4 EUCLIDEAN GEOMETRY


Euclidean space En
1.3
After these preparations, I am ready to introduce the main object of study: Euclidean
n-space (En , d) is a metric space (with metric d) for which there exists a bijective
map En ’ Rn , such that if P, Q ∈ En are mapped to x, y ∈ Rn then

d(P, Q) = |y ’ x|.

In other words, (En , d) is isometric to the vector space Rn with its usual distance
function, if you like this kind of language.
Since lines and collinearity in Rn are characterised purely in terms of the Euclidean
distance function, these notions carry over to En without any change: three points of
En are collinear if they are collinear for some isometry En ’ Rn (hence for all
possible isometries); the lines of En are the lines of Rn under any such identi¬cation.
For example, for points P, Q ∈ En , the line segment [P, Q] ‚ En is the set

[P, Q] = R ∈ En d(P, R) + d(R, Q) = d(P, Q) ‚ En .

The main point of the de¬nition of En is that the map En ’ Rn iden-
Remark
tifying the metrics is not ¬xed throughout the discussion; I only insist that one such
isometry should exist. A particular choice of identi¬cation preserving the metric is
referred to as a choice of (Euclidean) coordinates. Points of En will always be de-
noted by capital letters P, Q; once I choose a bijection, the points acquire coordinates
P = (x1 , . . . , xn ). In particular, any coordinate system distinguishes one point of En
as the origin (0, . . . , 0); however, different identi¬cations pick out different points of
En as their origin. If you want to have a Grand Mosque of Mecca or a Greenwich
Observatory, you must either receive it by Divine Grace or make a deliberate extra
choice. The idea of space ought to make sense without a coordinate system, but you
can always ¬x one if you like.
You can also look at this process from the opposite point of view. Going from Rn
to En , I forget the distinguished origin 0 ∈ Rn , the standard coordinate system, and
the vector space structure of Rn , remembering only the distance and properties that
can be derived from it.


1.4 Digression: shortest distance
As just shown, the metric of Euclidean space En determines the lines. This section
digresses to discuss the idea summarised in the well known clich´ ˜a straight line is
e
the shortest distance between two points™; while logically not absolutely essential in
this chapter, this idea is important in the philosophy of Euclidean geometry (as well
as spherical and hyperbolic geometry).

The distance d(P, Q) between two points P, Q ∈ En is the length of
Principle
the shortest curve joining P and Q. The line segment [P, Q] is the unique shortest
curve joining P, Q.
1.5 ANGLES 5


This looks obvious: if a curve C strays off the straight and narrow
Sketch proof
to some point R ∈ [P, Q], its length is at least
/

d(P, R) + d(R, Q) > d(P, Q).

The statement is, however, more subtle: for instance, it clearly does not make
sense without a de¬nition of a curve C and its length. A curve C in En from P to
Q is a family of points Rt ∈ En , depending on a ˜time variable™ t such that R0 = P
and R1 = Q. Clearly Rt should at least be a continuous function of t “ if you allow
instantaneous ˜teleporting™ between far away points, you can obviously get arbitrarily
short paths.
The proper de¬nition of curves and lengths of curves belongs to differential geom-
etry or analysis. Given a ˜suf¬ciently smooth™ curve, you can de¬ne its length as the
n
integral C ds along C of the in¬nitesimal arc length ds, given by ds 2 = i=1 dxi2 .
Alternatively, you can mark out successive points P = R0 , R1 , . . . , R N +1 = Q along
N
the curve, view the sum i=0 d(Ri , Ri+1 ) as an approximation to the length of C, and
de¬ne the length of C to be the supremum taken over all such piecewise linear ap-
proximations. To avoid the analytic details (which are not at all trivial!), I argue under
the following weak assumption: under any reasonable de¬nition of the length of C,
for any µ > 0, the curve C can be closely approximated by a piecewise linear path made up of
short intervals [P, R1 ], [R1 , R2 ], etc., such that

length of C ≥ sum of the lengths of the intervals ’ µ.

However, by the triangle inequality d(P, R2 ) ¤ d(P, R1 ) + d(R1 , R2 ), so that the
piecewise linear path can only get shorter if I omit R1 . Dealing likewise with R2 , R3 ,
etc., it follows that the length of C is ≥ d(P, Q) ’ µ. Since this is true for any µ > 0, it
follows that the length of C is ≥ d(P, Q). Thus the line interval [P, Q] joining P, Q
is the shortest path between them, and its length is d(P, Q) by de¬nition. QED


1.5 Angles
n
The geometric signi¬cance of the Euclidean inner product x · y = i=1 xi yi on Rn
(Section B.2) is that the inner product measures the size of the angle ∠xyz based at
y for x, y, z ∈ Rn :
(x ’ y) · (z ’ y)
cos(∠xyz) = . (8)
|x ’ y||z ’ y|
By convention, I usually choose the angle to be between 0 and π . In particular, the
vectors x ’ y, z ’ y are orthogonal if (x ’ y) · (z ’ y) = 0.
The notion of angle is easily transported to Euclidean space En . Namely, the angle
spanned by three points of En is de¬ned to be the corresponding angle in Rn under
a choice of coordinates. The angle is independent of this choice, because the inner
product in Rn is determined by the quadratic form (Proposition B.1), and so ultimately
6 EUCLIDEAN GEOMETRY


R




Q P
Figure 1.5 Angle with direction.



T “





Figure 1.6 Rigid body motion.


by the metric of En . In other words, the notion of angle is intrinsic to the geometry
of En .
There is one ¬nal issue to discuss regarding angles that is speci¬c to the Euclidean
plane E2 . Namely, once I ¬x a speci¬c coordinate system in E2 , angles ∠P Q R acquire
a direction as well as a size, once we agree (as we usually do) that an anticlockwise
angle counts as positive, and a clockwise angle as negative. In Figure 1.5,

∠P Q R = ’∠R Q P = θ.

Under this convention, angles lie between ’π and π . Of course formula (8) does not
reveal the sign as cos θ = cos(’θ). It is important to realise that the direction of the
angle is not intrinsic to E2 , since a different choice of coordinates may reverse the sign.


1.6 Motions
A motion T : En ’ En is a transformation that preserves distances; that is, T is
bijective, and

d(T (P), T (Q)) = d(P, Q) for all P, Q ∈ En .

The word motion is short for rigid body motion; it is alternatively called isometry or
congruence. To say that T preserves distances means that there is ˜no squashing or
bending™, hence the term rigid body motion; see Figure 1.6.
I study motions in terms of coordinates. After a choice of coordinates En ’ Rn , a
motion T gives rise to a map T : Rn ’ Rn , its coordinate expression, which satis¬es

|T (x) ’ T (y)| = |x ’ y| for all x, y ∈ Rn .
1.8 A MOTION IS AFFINE LINEAR ON LINES 7


The ¬rst thing I set out to do is to get from the abstract ˜preserves distance™ de¬nition of
a motion to the concrete coordinate expression T (x) = Ax + b with A an orthogonal
matrix. In the case of the Euclidean plane E2 , the result is even more concrete; A is
either a rotation matrix or a re¬‚ection matrix:
cos θ ’ sin θ cos θ sin θ
.
or
sin θ cos θ sin θ ’ cos θ



1.7 Motions and collinearity

A motion T : En ’ En preserves collinearity of points, so it takes
Proposition
lines to lines.

P, Q, R ∈ E n are collinear if and only if, possibly after a permutation of
Proof
P, Q, R,

d(P, R) + d(R, Q) = d(P, Q).

But T preserves the distance function, so this happens if and only if, possibly after a
permutation,

d(T (P), T (R)) + d(T (R), T (Q)) = d(T (P), T (Q))

which is equivalent to T (P), T (Q), T (R) collinear. QED
The point is of course that, as we saw in 1.3, collinearity can be de¬ned
purely in terms of distance; since a motion T preserves distance, it preserves
collinearity.


1.8 A motion is affine linear on lines

If T : Rn ’ Rn is a motion expressed in coordinates, then
Proposition

T ((1 ’ »)x + »y) = (1 ’ »)T (x) + »T (y)

for all x, y ∈ Rn and all » ∈ R.


A calculation based on the same idea as the previous proof: let z =
Proof
(1 ’ »)x + »y. If x = y there is nothing to prove; set d = |x ’ y|. Assume ¬rst that
» ∈ [0, 1], so that z ∈ [x, y]. Then, as in the previous proposition, T (z) ∈ [T (x), T (y)],
so T (z) = (1 ’ µ)T (x) + µT (y) for some µ. But |z ’ x| = »d, so T (z) is the point
at distance (1 ’ »)d from T (y) and »d from T (x), that is, µ = ».
If » < 0, say, then x ∈ [y, z] with x = (1 ’ » )y + » z and the same argument gives
T (x) = (1 ’ » )T (y) + » T (z), and you can derive the statement as an easy exercise.
(The point is to write » as a function of »; see Exercise 1.3.) QED
8 EUCLIDEAN GEOMETRY


1.9 Motions are affine transformations
A map T : En ’ En is an af¬ne transformation if it is given in a co-
Definition
ordinate system by T (x) = Ax + b, where A = (ai j ) is an n — n matrix with nonzero
determinant and b = (bi ) a vector; in more detail,
« « «
x1 x1 b1
n
¬.· ¬.· ¬.·
x = (xi ) ’ y = ai j x j + bi ,  .  ’ A .  +  . .
or (9)
. . .
j=1
xn xn bn

Let T : En ’ En be any map. Equivalent conditions:
Proposition

T is given in some coordinate system by T (x) = Ax + b for A an n — n matrix.
(1)
For all vectors x, y ∈ Rn and all », µ ∈ R we have
(2)

T »x + µy ’ T (0) = » T (x) ’ T (0) + µ T (y) ’ T (0) .

For all x, y ∈ Rn and all » ∈ R
(3)

T (1 ’ »)x + »y = (1 ’ »)T (x) + »T (y).

that is, T is af¬ne linear when restricted to any line.

The point of the proposition is that condition (3) is a priori much
Discussion
weaker than the other two; it only requires that the map T is af¬ne when restricted
to lines. Note also that using the origin 0 in (2) seems to go against my expressed
wisdom that there is no distinguished origin in the geometry of En . However, recall
that any point P ∈ En can serve as origin after a suitable translation.

(1) =’ (2) is an easy exercise. (2) means exactly that if after performing
Proof
T we translate by minus the vector b = T (0) to take T (0) back to 0, then T becomes
a linear map of vector spaces. Thus (2) =’ (1) comes from the standard result of
linear algebra expressing a linear map as a matrix.
(3) is just the particular case » + µ = 1 of (2). Thus the point of the proposition
is to prove (3) =’ (2).
Statement (2) concerns only the 2-dimensional vector subspace spanned by x, y ∈
V . We use statement (3) on the two lines 0x and 0y (see Figure 1.9), to get

T (2»x) = (1 ’ 2»)T (0) + 2»T (x)

and

T (2µy) = (1 ’ 2µ)T (0) + 2µT (y).

Now apply (3) again to the line spanned by 2»x and 2µy:
1.10 EUCLIDEAN MOTIONS AND ORTHOGONAL TRANSFORMATIONS 9




2»x



»x
»x + y
x



0 y y 2y

Affine linear construction of »x + µy.
Figure 1.9


1 1
T »x + µy = T (2»x) + T (2µy)
2 2
1 1
= (1 ’ 2»)T (0) + 2»T (x) + (1 ’ 2µ)T (0) + 2µT (y)
2 2
= T (0) + » T (x) ’ T (0) + µ T (y) ’ T (0) ,

as required. QED

Dividing by 2 here is just for the sake of an easy life: { 1 , 1 } is a conve-
Remark 22
nient solution of » + µ = 1. The point is just that »x + µy lies on a line containing
chosen points of 0x and 0y. The argument for (3) =’ (2) can be made to work
provided every line has ≥ 3 points, that is, over any ¬eld with > 2 elements.

A Euclidean motion T : En ’ En is an af¬ne transformation, given
Corollary
in any choice of coordinates En ’ Rn by T (x) = Ax + b.

This follows at once from Proposition 1.7, the implication (3) =’ (1) in the
previous proposition, and the fact that T is bijective, so the matrix A must be invertible.

1.10 Euclidean motions and orthogonal transformations
This section makes a brief use of the relationship between the standard quadratic
form |x|2 = xi2 on Rn and the associated inner product x · y = xi yi . If this is
not familiar to you, I refer you once again to Appendix B for a general discussion.

Let A be an n — n matrix and T : Rn ’ Rn the map de¬ned by
Proposition
x ’ Ax. Then the following are equivalent conditions:

T is a motion T : En ’ En .
(1)
A preserves the quadratic form; that is, |Ax| = |x| for all x ∈ Rn .
(2)
A is an orthogonal matrix; that is, it satis¬es tA A = In .
(3)
10 EUCLIDEAN GEOMETRY


(1) =’ (2) is trivial. Conversely,
Proof

|Ax ’ Ay|2 = |A(x ’ y)|2 = |x ’ y|2 ,

where the ¬rst equality is linearity, and the second follows from (2). Thus T preserves
length, so it is a motion. (2) ⇐’ (3) is proved in Proposition B.4, where you can
also read more about orthogonal matrixes if you wish to. QED
Together with Corollary 1.7, this proves the following very important statement:

A Euclidean motion T : En ’ En is expressed in coordinates as
Corollary

T (x) = Ax + b

with A an orthogonal matrix, and b ∈ Rn a vector.
An immediate check shows that an orthogonal matrix A has determinant det A =
±1 (see Lemma B.4).

Let T : En ’ En be a motion expressed in coordinates as T (x) =
Definition
Ax + b. I call T direct (or orientation preserving) if det A = 1 and opposite (or
orientation reversing) if det A = ’1.

The meaning of this notion in E2 and E3 is familiar in terms of left“right orientation,
and it may seem pretty intuitive that it does not depend on the choice of coordinates.
However, I leave the proof to Exercise 6.8.


1.11 Normal form of an orthogonal matrix
The point of this section is to express an orthogonal map ± : Rn ’ Rn in a simple form
in a suitable orthonormal basis of Rn . This section may seem an obscure digression
into linear algebra, but the result is central to understanding motions of Euclidean
space.

As a prelude to an attack on the general problem, consider the instructive case n = 2.
1.11.1
The 2 — 2 The conditions for a 2 — 2 matrix A = a d to be orthogonal are:
b
c
rotation and ±
 a 2 + c2 = 1

reflection
ac ab 10
A A = 1 ⇐’ = ⇐’ ab + cd = 0
t
matrixes
2
bd cd 01 
b + d 2 = 1.

Now (a, c) ∈ R2 is a point of the unit circle, so I can write a = cos θ, c = sin θ
for some θ ∈ [0, 2π) (Figure 1.11a). Then there are just two possibilities for b, d,
giving

cos θ ’ sin θ cos θ sin θ
A= .
or
sin θ cos θ sin θ ’ cos θ
1.11 NORMAL FORM OF AN ORTHOGONAL MATRIX 11




(0, 1)
(cos θ, sin θ)


(’ sin θ, cos θ)
θ

θ

(1, 0)
Figure 1.11a A rotation in coordinates.










Refl(L)
L
Rot(O, θ)

θ/2
y=0
θ/2
θ “
O






Figure 1.11b The rotation and the reflection.


The ¬rst of these corresponds to a direct motion (because det A = 1), and you
recognise it as a rotation around the origin through θ. In fact it takes

cos θ ’ sin θ
1 0
’ ’ .
and
sin θ cos θ
0 1

The second matrix gives an opposite motion (det A = ’1), and you can understand
it in several ways; for example, write

cos θ sin θ cos θ ’ sin θ 1 0
A= = .
sin θ ’ cos θ sin θ cos θ ’1
0

This says: ¬rst re¬‚ect in the x-axis, then rotate through θ. It is easy to see geo-
metrically that this is the re¬‚ection in the line L through the origin 0 at angle θ/2
to the x-axis. Indeed, every point on L is ¬xed, and the line perpendicular to L is
reversed, as in Figure 1.11b.
In coordinates, this says that f1 = (cos(θ/2), sin(θ/2)) is an eigenvector of A with
eigenvalue 1, and f2 = (sin(θ/2), ’ cos(θ/2)) an eigenvector with eigenvalue ’1.
The pair (f1 , f2 ) gives a vector space basis of R2 , and in this new basis the map
is given by the matrix 1 ’1 . You can readily check these statements by matrix
0
0
multiplication and the rules of trig, but the geometric argument is simpler and more
convincing.
12 EUCLIDEAN GEOMETRY


1.11.2 The In the general case I control orthogonal matrixes using a slightly more involved
general case argument.

Let ± : Rn ’ Rn be a linear map
Theorem (Normal form of orthogonal matrix)
given by an orthogonal matrix A. Then there exists an orthonormal basis of Rn in
which the matrix of ± is
« 
Ik +
¬ ·
¬ ·
’Ik ’
¬ ·
cos θi ’ sin θi
¬ ·
B=¬ Bi = .
· where
B1
¬ · sin θi cos θi
¬ ·
..
 
.
Bl

Here k + + k ’ + 2l = n, and Ik ± is the k ± — k ± identity matrix.

cos θ ’ sin θ
has two special cases θ = 0 (giving
The rotation matrix
Discussion sin θ cos θ
the identity) and θ = π:

’1 cos π ’ sin π
0
= 180—¦ rotation.
=
’1 sin π cos π
0

These trivial cases introduce a minor ambiguity in the normal form. The most natural
convention seems to be to disallow θ = 0, thus taking k + as big as possible, but to
use θ = π wherever possible, so that k ’ = 0 or 1.

In sketch form, this holds because A is orthogonal, so its eigenvalues
Proof
have absolute value 1. Therefore they are either ±1, or come in complex conjugate
pairs {», »} = exp(±iθ); after this, it is enough simply to build up a basis of Rn
consisting either of real eigenvectors of A, or of real and imaginary parts of complex
eigenvectors.
Now I say the same thing again in more detail in 5 steps; the sketch proof just
given already reveals that complex numbers are closely involved, so I may as well
extend the action of A to the complex vector space Cn , which I can do without any
problems.

If » is a real eigenvalue of A then » = ±1, because
Step 1

Ax = »x and A orthogonal =’ |x|2 = |Ax|2 = »2 |x|2 .

If » is a complex eigenvalue of A then |»| = 1 and » = »’1 is also an
Step 2
eigenvalue (the bar denotes complex conjugate). Indeed, given 0 = z ∈ Cn such that
Az = »z (recall I write z = t(z 1 , . . . , z n ) a column vector), write z = t(z 1 , . . . , z n ).
1.11 NORMAL FORM OF AN ORTHOGONAL MATRIX 13


Because A is a real matrix,

Az = Az = »z = »z.

Now write z i = xi + iyi , so that t zz = |z i |2 = (xi2 + yi2 ) > 0. Using the fact
that A is orthogonal,

»»t zz = t(Az)Az = t z t A Az = t zz, »» = 1.
and thus


If » = cos θ + i sin θ is a complex eigenvalue of A (with θ = 0, π ) and
Step 3
z = x + iy ∈ Cn a complex eigenvector then taking real and imaginary parts in the
equality A(x + iy) = Az = »z = (cos θ + i sin θ)(x + iy) gives

Ax = cos θx ’ sin θy, Ay = sin θx + cos θy. (10)

Now I claim that |x|2 = |y|2 and x · y = 0, so that scaling makes x, y ∈ Rn into a pair
of orthonormal vectors. This is an exercise for the reader. [Hint: write out the condition
for (10) (with θ = 0, π ) to preserve |x|2 , |y|2 and x · y. See Exercises 1.5“1.6.]


If ± preserves a subspace W of Rn , then it preserves its orthogonal com-
Step 4
plement under the inner product (compare B.3)

W ⊥ = x ∈ Rn x · w = 0 for all w ∈ W .

In symbols,

±(W ) = W =’ ±(W ⊥ ) = W ⊥ .

This is obvious from the de¬nition of W ⊥ . Look at Figure 1.15b for an example: if a
motion preserves the horizontal plane W and its translates, then it will also preserve
the orthogonal complement W ⊥ , the vertical lines.


Eigenvalues of A come from the polynomial equa-

. 1
( 8)



>>