Geometry of matrices : in memory of Professor L.K. Hua (1910-1985)

GEOMETRY

Of: MATRICES

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GEOMETRY OF MATRICES In Memory of Professor L K Hua (1910 -1985)

Zhe-Xian Wan Chinese Academy of Sciences, China Lund University, Sweden

{World Scientific Singapore'New Jersey • L Singapore* NewJersey•London •Hong Kong

Published by World Scientific Publishing Co Pte Ltd P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Wan, Che-hsien. Geometry of matrices / Zhe-xian Wan. p. cm. "In memory of Professor L. K. Hua (1910-1985)." Includes bibliographical references and index. ISBN 9810226381 1. Matrices. I. Title. QA188.W36 1996 516.3'5--dc20

96-2179 CIP

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Preface The present monograph is a state of the art survey of the geometry of matri­ ces. Professor L. K. Hua initiated the work in this area in the middle forties. In this geometry, the points of the space are a certain kind of matrices of a given size, and the four kinds of matrices studied by Hua are rectangular matrices, symmetric matrices, skew-symmetric matrices and hermitian ma­ trices. To each such space there is associated a group of motions, and the aim of the study is then to characterize the group of motions in the space by as few geometric invariants as possible. At first, Professor Hua, relating to his study of the theory of functions of several complex variables, began studying the geometry of matrices of various types over the complex field. Later, he extended his results to the case when the basic field is not necessar­ ily commutative, discovered that the invariant "adjacency" alone is sufficient to characterize the group of motions of the space, and applied his results to some problems in algebra and geometry. Professor Hua's pioneer work in the area has been followed by many mathematicians, and more general results have been obtained. I think it is now time to summarize all results obtained so far, and this has been my motivation for the present work. In order to be as self-contained as possible the book covers some material of linear algebra over division rings in Chapter 1, which is necessary for later chapters. This chapter can also be read independently as an introduction to linear algebra over division rings. The fundamental theorems of the affine geometry and of the projective geometry over any division ring constitute the main contents of Chapter 2. In particular, Hua's beautiful theorem on semi-automorphisms of a division ring and its application to the fundamental theorem of the one-dimensional projective geometry over a division ring are v

PREFACE

VI

included. Following these chapters, the geometry of rectangular matrices over any division ring, alternate matrices over any field, symmetric matrices over any field, and the geometry of hermitian matrices over any division ring which possesses an involution are discussed in detail in Chapters 3, 4, 5, and 6, respectively. Applications to problems in algebra, geometry, and graph theory are included throughout. Finally, the author is indebted to Yangxian Wang and Mulan Liu for their helpful comments on the first draft of the book, to Rongquan Feng, Lei Hu, Xinwen Wu, and Zhanfei Zhou for their laborious typewriting, and to Lena Mansson for her beautiful improvement of the camera-ready copy. Zhe-xian Wan

Contents Preface

v

1

Linear Algebra over Division Rings 1.1 Vector Spaces over Division Rings 1.2 Matrices over Division Rings 1.3 Matrix Representations of Subspaces 1.4 Systems of Linear Equations 1.5 Hermitian, Symmetric, and Alternate Matrices 1.6 Comments

1 1 11 27 29 35 44

2

Affine G e o m e t r y and Projective G e o m e t r y 2.1 Affine Spaces and Affine Groups

45 45

2.2 2.3 2.4 2.5 2.6

54 66 76 80 87

3

Fundamental Theorem of the Affine Geometry Projective Spaces and Projective Groups Fundamental Theorem of the Projective Geometry One-dimensional Projective Geometry Comments

G e o m e t r y of Rectangular Matrices 3.1 The Space of Rectangular Matrices 3.2 Maximal Sets of Rank 1 3.3 Maximal Sets of Rank 2 3.4 Proof of the Fundamental Theorem 3.5 Application to Algebra 3.6 Application to Geometry 3.7 Application to Geometry (Continued) vn

89 89 93 97 106 118 123 139

viii

CONTENTS 3.8 3.9

Application to Graph Theory Comments

153 155

4

G e o m e t r y of Alternate Matrices 4.1 The Space of Alternate Matrices 4.2 Maximal Sets 4.3 Proof of the Fundamental Theorem 4.4 Application to Geometry 4.5 Application to Geometry (Continued) 4.6 Application to Graph Theory 4.7 Comments

157 157 159 168 177 199 211 215

5

G e o m e t r y of Symmetric M a t r i c e s 5.1 The Space of Symmetric Matrices 5.2 Maximal Sets of Rank 1 5.3 Maximal Sets of Rank 2 (Characteristic Not Two) 5.4 Proof of the Fundamental Theorem (I) 5.5 Maximal Sets of Rank 2 (Characteristic Two) 5.6 Proof of the Fundamental Theorem (II) 5.7 Proof of the Fundamental Theorem (III) 5.8 Application to Algebra 5.9 Application to Geometry 5.10 Application to Graph Theory 5.11 Comments

217 217 222 224 231 244 252 264 281 285 296 303

6

G e o m e t r y of Hermitian Matrices 6.1 The Space of Hermitian Matrices 6.2 Maximal Sets of Rank 1 6.3 Maximal Sets of Rank 2 6.4 Proof of the Fundamental Theorem (the Case n > 3) . . . . 6.5 Maximal Sets of Rank 2 (the Case n = 2) 6.6 Proof of the Fundamental Theorem (the Case n = 2) . . . . 6.7 Application to Algebra 6.8 Application to Geometry 6.9 Application to Graph Theory

305 305 308 311 323 341 348 355 356 363

CONTENTS 6.10 Comments

ix 365

Bibliography

367

Index

371

Chapter 1 Linear Algebra over Division Rings 1.1

Vector Spaces over Division Rings

Let D be any division ring and n a positive integer. We use D{n) = {On, a?2, • • •, i n ) I Xi G D, i = 1, 2, • • •, n} to denote the n-dimensional row vector space (or left vector space) over D formed by the set of all n-tuples (or n-dimensional row vectors) (xu

x 2 , • • •, x n ),

Xi£ D, i = 1, 2, • • •, n,

over D with addition and scalar multiplication defined by (xu x 2 , • • •, xn) + (2/1,2/2, * • •, 2/n) = («i + 2/1, x2 + 2/2, • • • 9xn + yn) and X\Xiy

X 2 , * " * , Xfij

— y*LXi)

aJX 2 ,

,

d/Xjij^

respectively, where x, xi, ar2, •••, xn, ?/i, ?/2, •••, Vn € D. For any row vectors u, u, iu G i ^ and re, y G D we have the following manipulation rules U + V = V + U,

(u + v) + w = u + (v + w), x(?i + v) = xu + xv, (x + 2/)^ = a;w + yu, (xy)u = x(yu), lu = u, 1

2

Chapter 1. Linear Algebra over Division Rings

where 1 is the identity element of D. Moreover, let dci = = (l,0,0,-( l , 0 , 0 , - - -•,0), , 0 ) , ee22 = = (0,1,0,-• (0,l,(),•••,0), ( 0 , 0 , - -•,o,i), -,0,l), •,o), •••••,•, ee nn == (0,0,-then ei, e 2 , •••, e n form a basis of D^n\ i.e., any vector (#i, £ 2 , •••, # n ) of D^ can be written as a linear combination of ei, e 2 , • • •, e n with coefficients in D: (xi, xZ22,, •' • • •, •, xxnn)) = = xiei xnen, xxex + + zx22ee22 H + • •~r\~ ^n^nt 0*1, and the expression is unique. Let v = (ai, a 2 , • • •, a n ) be an n-dimensional row vector of D^n\ then a t (1 < i < n) is called the i-th component of v. If all the components of v are 0, i.e., v=(0, 0, • • •, 0), then v is called the zero (row) vector and denoted simply by 0, i.e., (0, 0, 0)==0.0. (o, o, • •■•,•, 0) Clearly, for any row vector v € D^

we have

V. v+ 0= == 0 + v === v.

Let v = (ai, a 2 , • • •, an) G D^

and define

-v -v = = (-ai, (—ai, --aa 22 ,, •••, • •» • —a - f l nn)),, then vw + + ((-v) 0. --v)-- == 0. Definition 1.1: Let t>i, v 2 , • • •, vT be r row vectors in D^1'. They are said to be linearly dependent (over D) if there are r elements ai, a 2 , • • •, ar of Z), which are not all equal to 0, such that a\Vi + a2v22 + ai^i + •-••h a+rvaTrvr= = 0; 0; otherwise, they are said to be linearly independent

□

It follows immediately from Definition 1.1 that any finite number of row vectors in , among which the zero vector 0 appears, are linearly depen­ dent and that the vectors of any nonempty subset of a finite set of linearly independent vectors are linearly independent.

1.1. Vector Spaces over Division Rings

3

Definition 1.2: Let v, ui, v2, • • •, vr be row vectors in Z)(n). If there are elements a1? a 2 , • • ■, a r G JD such that v = ai^i + a 2 u 2 H

f- arvr,

then we say that v is a linear combination of vi, u2, • • •, v r with coefficients «i, a 2 , • • •, a r in -D.

D

Let vi, u 2 , • • •, vr be r row vectors in 2 ? ^ , denote by < v x , u2, • • •, v r > the set of all linear combinations of v\, u2, • • •, vr which coefficients in D. Clearly, if u, v G < Vi, t>2, • • •, vr > and a £ D, then u + u G f l and av G 2). More generally, we have Definition 1.3: A nonempty set V of D^ with the property that u , u G V and a G D imply u + v G V and av G D is called a subspace of 2}(n). If V consists only of the zero vector, then V is called the zero subspace and denoted by < 0 >; otherwise, V is called a nonzero subspace. □ For example, < vi, v2, • • •, v r > is a subspace of D^ a subspace of D^.

and D^

itself is also

Definition 1.4: Let V be a subspace of D ' n ' . If there are row vectors vi, v2, • • •, vr such that V = < v\, u2, * • •, vr >, then they are said to form a set of generators of V and V is said to be spanned by them. Furthermore, if v1? u2?* • •, v r are linearly independent, then they are said to form a basis of V.

□ For example, ei, e 2 , • • •, e n form a basis of 2)(n). We want to prove that every subspace of D^ has a basis and the numbers of elements in different bases of a subspace are the same and < n. We proceed as follows. L e m m a 1.1: Letvi, v 2 , •••, vr be row vectors in D^. Thenwi, t>2, •••, vr are linearly independent if and only if any row vector v G < vi, v2, • • •, vr > can be expressed uniquely as a linear combination of t>i, v 2 , • • •, vr. Proof: Assume that vi, v2, • • •, vr are linearly independent. Let v be any vector of < Vi, u2, • • •, vr >. If v has two representations as linear combinations of vi, v2, • • •, wr, say u = a\V\ + a2V2 + • • • + a r v r

4

Chapter 1. Linear Algebra over Division Rings

and vr, v = 622vv22 H+ • -• • +h b£Tw = biVi 61^2 + b where a t , 6t G 2? (z = 1,2, • • • , r ) , then subtracting the second expression from the first one, we obtain {arr--0 = (en (ai -- h)vi 6i)vi + (a 2 - 62)v2 + • -• +h (a

br)vr.

Since ui, v2, ***> v r are linearly independent, we have at- = 6t-

(z =

1, 2, • ■ ■ , r ) .

Conversely, suppose that any row vector v G < Vi, v 2 , • • •, vr > can be expressed uniquely as a linear combination of vi, v 2 , • • •, vr. Let ai, a 2 , • • •, ar £ D be such that aiVi • +Gr^V arvr ==0.0. ai^i + a22vv22 + • •■■• +

We also have 0vx1 + 0v2 + •"'■ + • +0v0v r r==0.0. By the uniqueness of the zero vector 0 as a linear combination of v\, v2, • • •, v r , we have a t = 0 (z = 1, 2, • • •, r). Therefore vi, v2, • • •, u r are linearly independent. D L e m m a 1.2: If ui, w2, • • •, us G < i>i, v2, • • -, vr > and u> G < ui, u 2 , • • •, u 5 >, then w G < vi, v2, • • •, v r >. Proof: From

rr

(t " , 5*' s)) (* == l,*> 22,> •••*

i = ]Cfl«*V*

U

4=1 A:=l

and s

w ii = = ll

where alfc, 6t- € 25 (i = 1, 2, • • •, 3; k = 1, 2, • • •, r), we deduce s s

r

r

s

kVk = U> = X) b* 2 3 a^iiJbVJb ^2(YJ W

*i=l =1

k=l

k=l

i'=l

h a

i*)v*. i ik)VkD

Theorem 1.3 (Replacement Theorem): Let ?ii, u2, •••, u s be 5 linearly independent elements of the subspace < vi, v 2 , • • •, vr >, then there exist 5

5

1.1. Vector Spaces over Division Rings

elements vt-a, vt-2, • • •, vis (1 < z'i < z2 < • • • < is < r) in {^i, v 2 , • * *, vr} such that if we replace them by u i , W2, • • •, us in {vi, v2, • • •, ^ r } , the resulting set is also a system of generators of < u1? u2, • • •, v r > . In particular, s < r. Proof: We prove this theorem by induction on 5. If 5 = 0, the theorem holds trivially. Assume now that the theorem holds for 5 — 1. Then there exist 6 — 1 elements t ^ , ut-2, • • •, V{s_l (1 < ix < i2 < • • • < zs_i < r) in {vi, v2, • • •, v r } such that if they are replaced by ui, u 2 , • • •, w s -i then the resulting set {ui, u 2 , * • *, ^ s - i , ut-tf, v ts+1 , • • •, vt-r} is also a system of generators of < Vi, v 2 , • • •, v r > , where z'i, permutation of 1, 2, • • •, r. Since ws G< vi, ^25 • • •, t>r > < vi,v v1,v22,,-',v-r '
= < u u ,-

an

d - ir,v> ir ,> , •-■,v

• •, W ,u—1 ,, ^t' s.uv,isf,vis+1

, Vr > = < Uui , 2 W2,- •

S

?s

5 " "

s+ 1

we have = aa^ui + aa22uu2 2 + -\ • •■• +h a ss _it/ _ i t / s _i _i + + a assV{ v;s5 + + a a ss+1 vis+1 + • •• ■ uus = i^i + + iV z - 5 + 1 -\

(1.1) +\- aarrvvirir,, (1.1)

where ai, a2, • • •, a s _i, a s , a s + i , • • •, ar G D. By hypothesis t/i, U2, • • •, us are linearly independent over D, thus a s ,a s +i, • • •, ar cannot all be 0. Without loss of generality we assume that the subscripts zs, z s +i, • • •, ir have been so chosen that as ^ 0. Then •— Vi - a / a aiUi i ^ i -—• •• •• •——asa~asa-iu asajusus-a— a~ • a• s • —arV{ a~r. arV{r. s-iu s-is-i ++ s+iVis+1• •— s a s+1vais+1 is s = = —aj V

It follows that V{3 G < wi, • • •, w s _i, u s , ut-a+1, • • •, V{r > and consequently
C < ui,--

• , Us—1, U s , ^z' s+1 5 ' '

• , ^tv > •

On the other hand it follows from (1.1) that us G < t*i, u 2 , • • •, u s - i , ^t-a, u

t a+ i?

,,,

, U t r > and therefore

< u iUl," , - - - ,, uWs __ ii ,, uU s ,, ^^ i5 + 1 , - - --,v, ir^ r > >C C< >v < vuv2,- 'r>,V

r

= < uUuui 2,,-U , "- ' ,u ,via+1 ,vir >> 5 V's-l') ) Vi,- , • •• • •, Vir 8-i,viaV{ = < U-L , • • •, w -i, w , v; , • • vir .> . • , U —1, 1/ , ^z' • * •, v = < w i , w, w , s s s+1 ir•, > 2 2

> =
, then V\ forms a basis of V. If V ^ < V\ > , then there is an element v2 in V such that v2 # < v\ >. Thus < Vi,v2 > Q V and vx, v2 are linearly independent. If V = < v\,v2 >, then Wi, v2 form a basis of V. If V =fi < v^v2 >, then there is an element v 3 in V such that v3 £ < vi,v2 >. Thus < vi,t>2,^3 > Q V a n d v i^ v 2, ^3 are linearly independent. By the Replacement Theorem the number of elements in any set of linearly independent vectors of D^ = < ei, e 2 , • • •, e n > is < n. Thus proceeding in the above way, we will arrive at a set of linearly independent elements vi, v2, • • •, vr in V such that V = < t>i, v2, • • •, vr > and r < n. Hence i>i, v2, • • •, v r form a basis of V. This proves that V has a basis whose number of elements is < n. Let {vi,^2,-• • ,v r } and {1/1,1*2, • • •,u s } be two bases of V, then V = < ^ i ^ 2 , • • •, vr > = < tii, ^2, • • •, us > . By the Replacement Theorem we have r < s and s < r. Therefore r = s. □ Based on Proposition 1.4 we give the following definition. Definition 1.5: The number of elements in any basis of a nonzero subspace V of D^ is called the dimension of V and is denoted by dim V. In particular, dim Z)(n) = n. The dimension of the zero subspace of D^ is defined to be 0, i.e., d i m < 0 > = 0. D Corollary 1.5: The dimension of < v1? v2, • • •, vr > is the maximal number of linear independent elements among v1? v2j • • •, vr. D Proposition 1.6: Let V and W be subspaces of D^ of dimensions / and m, respectively. Assume that V CW. Then / < m and to any basis of V we can add m — / elements of W such that the resulting set is a basis of W. In particular, if V C W and / = m, then V — W. Proof: It follows immediately from Theorem 1.3.

□

1.1. Vector Spaces over Division Rings Now let us study the intersection and sum of subspaces of

7 D^.

Proposition 1.7: Let V\ and V2 be two subspaces of D^n\ Then the set row vectors belonging to both V\ and V2 is a subspace of D^ and the set row vectors which can be written as sums of a vector of V\ and a vector V2 is also a subspace of D^.

of of of □

The proof is easy and hence omitted. Definition 1.6: Let Vi and V2 be two subspaces of D^n\ The subspace of Z)(n) which consists of those row vectors belonging to both V\ and V2 is called the intersection of Vi and V2 and denoted by V\ f) V2. The subspace of Z)(n) consisting of those row vectors which can be written as sums of a vector of V\ and a vector of V2 is called the sum of Vi and V2 and denoted by Vi + V2. □ The following properties of the intersection and sum of subspaces are imme­ diate. Let V, Vi, V2, and V3 be subspaces of D^n\ then

vnv = v, v + v = V; v1nv2 = v2nVi, vl + v2 = v2 + Vi; vi n {v2 n v3) = (Vi n v2) n v3, v1 + (v2 + v3) = (vi + v2) + v3; vi n (Vi + v2) = Vi, vi + (Vi n v2) = vi. More importantly, we have Proposition 1.8 (Dimension Formula): Let Vi and V2 be subspaces of D^n\ then dim VL + dim V2 = dim (Vi C\V2) + dim (Vi + V2). Proof: Let dim(Vi fl V2) = d and ui, v2, • • •, ^ be a basis of Vi D V2. Clearly Vi fl V2 C Vi and Vi n V2 C F 2 . Let dim Vi = s and dim V2 = t. By Proposition 1.6, d < s, d < t, there exist elements ui, u 2 , • • •, u3-d in Vi such that v\, v2, • •, vj, u 1? w2, • • •, u5_d form a basis of Vi, and there exist elements wi, w2, • • •, wt-d in V2 such that vi, v2, • • •, Vd, u>i, w2, • • •, wt-d form a basis of V2. Clearly V1 + V2 = .

If we can prove that vi, • • •, vj, «i, • • •, wa_d, itfi, • • •, wt-d are linearly inde­ pendent, then our theorem will be proved. Suppose that there are elements

8

Chapter 1. Linear Algebra over Division Rings

0i) * * •) ad, h, ' • •, ^5-d, ci, * • ■? Ct-d G D such that aiui H

h a^v* + ftitii H

h bs-dus-d

+ ciwi H

h adVd + 6iMi H

h bs-dus-d

= —(ciWi H

h ct-dwt-d

- 0.

Then ai^i H

h

ct-dwt-d).

The left-hand side of the above equality is an element of Vi and the righthand side is an element of V2, hence it is an element of Vi fl V2. It follows that 61 = • • • = bs-d = 0 and c\ = • • • = ct-d = 0 and consequently are ai = ••• = ad = 0. Therefore t>i, • • •, v^, uu • •, u5_d, Wi, • • •, wt-d linearly independent. D From the dimension formula we deduce Proposition 1.9 (Modular Law): Let Vi, V2, and V3 be subspaces of and assume that Vi C V3. Then

D^

K + (v2nv3) = {K + v2) nv3. Proof: From Vi C Vi + V2 and Vi C V3 we deduce Vi C (Vi + V2) n V3. Clearly, V2 0 F 3 C (Vi + V2) n V3. Therefore Vi + (V2 H F3) C (Vi + V2) n V3. If we can prove

dim(Vi + (v2 n y3)) = dim((Vi + v2) n v3), then our proposition follows immediately from Proposition 1.6. By Propo­ sition 1.8, we have dim(Vi '+ (V2 H V3)) = dim Vi + dim( V2 n V3) - dim(Vi f)V2n V3) = dim Vi + dim(y 2 H V3) - dim(Vi fl V2) (as Vi C V3) = dim Vi + dim V2 + dim V3 — dim( V2 + V3) - ( d i m Vi + dim V2 - dim(Vi + V2)) = dim V3 + dim(Vi + V2) - dim(V2 + V3) and dim((Vi + V2) H V3) = dim(Vi + V2) + dim V3 - dim(Vi + V2 + V3) = dim V3 + dim(Vi + V2) - dim(T/2 + V3) (as Vi C V&). D

9

1.1. Vector Spaces over Division Rings

We remark that if the condition v\ C V3 does not hold, then the conclusion of Proposition 1.9 may not be true. It is not difficult to give counter-examples. Proposition 1.10: For any subspace V of D^ subspace W such that D^

and

= V +W

VHW

=

there exists at least a

.

Proof: Let dim V = r and v l5 u2, • • •, vT be a basis of V. By Proposition 1.6 there are row vectors i/i, u 2 , • • *, un-r such that vi, v 2 , • • •, ^r, ^ i , ^2, • • •, un-r form a basis of D^n\ Let W = < i/i, u 2 , • • ■, u n -r >, then

£>(") = y + w and v n w = < o > . Correspondingly, we have the n-dimensional column vector space (or right vector space) over JD, which consists of all n-dimensional column vectors

.

,

X{, e D, i = 1, 2, • • •, n,

\ Zn /

over D with addition and scalar multiplication defined by / 2/1 \ 2/2

^2

/ ^1 + y\ \ I ^ 2 + 2/2

+ \ ^n /

V 2/n /

\ Xn + Vn /

and ( X\X

\

X2X

\ Xn /

\ xnx I

respectively, where #, xi, # 2 , • • •, xn, j/i, y 2 , • • •, yn G D. We use the symbol *D(n) to denote the n-dimensional column vector space over D and lx to denote a vector in it, where x is an n-dimensional row vector over D. We have analogous results for *Z?(n) but they will not be repeated here.

10

Chapter 1. Linear Algebra over Division Rings

Finally, let us give the definition of an (abstract) vector space over a division ring. Definition 1.7: Let D be a division ring and V a nonempty set. Assume that to any two elements v and w of V there corresponds uniquely an element of V, which is called the sum of v and w and denoted by v + w. This correspondence is called the addition of V. Next assume that to any element A £ D and v G V there corresponds uniquely an element of V which is called the scalar product of v by A from the left and denoted by Xv. This correspondence is called the scalar multiplication of V. We say that V is a left vector space over D if the following manipulation rules I and H hold in V. I

V is an abelian group with respect to the addition.

H For any A, fi £ D and v, w € V, we have (i) (ii) (iii) (iv)

X(v + w) = Xv + Xw, (A + fi)v = Xv + fiv, (\p)v = A(/*v), lu = u.

We call the elements of V vectors, the zero element of the abelian group with respect to addition the zero vector, denoted by 0, and the elements of D scalars. Furthermore, besides I and H, assume that the following also holds: 1H There are n vectors ei, e 2 , • • •, e n of V such that any vector v can be expressed uniquely as their linear combination v = cici + c 2 e 2 + • • • + c n e n , with coefficients Ci, c 2 , • • •, cn in D. Then V is called a finite dimensional left vector space over D, n its dimen­ sion, and {ei, e 2 , • • •, e n } a basis of V. □ is an example of a finite dimensional left vector space over D of dimen­ sion n.

11

1.2. Matri ces over Di vision Rings

Two left vector spaces over the same division ring are said to be isomorphic, if there is a bijective map from one to the other which preserves the addition and scalar multiplication. Parallel to the discussion of D^n\ for finite dimensional left vector space V over a division ring D we can repeat all the definitions and propositions given in this section, but the details will be omitted. Similarly, we can also define finite dimensional right vector spaces over D. When D = F is a field, the map from F^

to

f

F^

I «i \ a2

(ai, a 2 , •••, an)

(1.2)

\ «» / is bijective and preserves the addition and scalar multiplication of vectors. In fact, (ai,a 2 ,- • • , a n ) + (61,62,- • ■, 6n) = («i + &i>a2 + 62, * •• , a n + M / ai + 61 \ a 2 + 62

/ d\ \ a2

V a n + bn }

\ «n /

/ 6l \ 62

+

VK )

and a(ai, a 2 , • • •, a n ) = (a«i, aa 2 , • • •, aan) ( ««i

aa 2

\

/ aia \ a a

2

( «i

\

«2

\ ^ / Thus F^ ^ and F^ can be regarded as isomorphic. Usually we identify and 2n

Q"mm /

and

{

\ #33

#34

#43

#44

/

13

1.2. Matrices over Division Rings respectively. Let us introduce the operations on matrices. (i) The addition of matrices. Let A = {ij)imn) )

It can be easily proved that

'{'A) --=A,A, *('A) t

l

(A ++ 5B)) = A + *B,

\

vn

)

1.2. Matrices over Division Rings

23

Then ' A B is invertible.

□

Definition 1.9: Let A and B be both m x n matrices over a division ring D. They are said to be equivalent if there is an m x m invertible matrix P and an n x n invertible matrix Q such that A = PBQ. □ Proposition 1.16: Equivalent matrices have the same row rank and the same column rank. Proof: Let A and B be equivalent m x n matrices over a division ring D. That is, there is an m x m invertible matrix P and an n x n invertible matrix Q such that A = PBQ. Then P " 1 A = BQ. By Proposition 1.14 B and BQ have the same row rank. Clearly A and P~lA have the same row rank. Therefore A and B have the same row rank. Similarly we can prove that A and B have the same column rank.

□

Proposition 1.17: Any m x n matrix of row rank r over a division ring is equivalent to I(r) r\(m—r,n—r)

Proof: Let A be an m x n matrix of row rank r over a division ring D. An mxm matrix P is called a permutation matrix if every row and every column of P has only one 1 and m — 1 O's. Clearly lPP — / , hence permutation matrices are invertible matrices. Moreover, PA is a matrix obtained from A by permuting its rows. Since A is of row rank r, we may choose an mxm permutation matrix P such that the first r row of PA are linearly independent, then the last m — r rows of PA are linear combinations of the first r rows. Write

PA

v2 \Vrn/

Chapter 1. Linear Algebra over Division Rings

24 then

r

i = r + l,r + 2,-- • ,m.

i=i

Let

Band

= (bij)r +l

f VVl ) 2

\VT

)

Since R is of row rank r, by Proposition 1.15 there is an (n — r) x r matrix D such that (' R R\ • _ ._

Q=

is an n x n invertible matrix. Then

( /W I

-i r

-1

[»)

Q(m-r,n-r)

\^

J r. Proof: Since A is of rank r, A is of row rank r and there are r linearly independent rows of A. Denote the submatrix formed by these r linearly independent rows of A by A\. Then A\ is an r x n matrix of rank r. Hence A\ is also of column rank r and there are r linearly independent columns of A\. Denote the submatrix formed by these r linearly independent columns of Ai by A2. Then A2 is an r x r submatrix of A\ and also of A. A 2 is of column rank r and, hence, rank r. Let B be an s x s submatrix of A and s > r. If B is invertible, by Proposition 1.11 the s rows of B are linearly independent. It follows that the s rows of A at which the s rows of B are situated are also linearly independent. Then rank A > 5, which is a contradiction. □ Now we give two propositions on the bounds of the ranks of the sum and product of two matrices. P r o p o s i t i o n 1.21: Let A and B both be m x n matrices over a division ring D. Then rank(A + B) < rank A + rank B. Proof: By Proposition 1.16 and Corollary 1.18 the rank of a matrix is invariant under the equivalence transformation X 1—* PXQ, where X is an m x n matrix, P G GLm(D), and Q G GLn(D). By Proposition 1.17, without loss of generality we can assume that A =

JM

\ r\(m—r,n—r)

where r = rank A. Let

*-(£)•

Ii

26

Chapter 1. Linear Algebra over Division Rings

where B\ is an r x n matrix and B2 is (m — r) x n. Then rank(A + B) < rank((J( r ) 0) + Bx) + rank B2 < rank A + rank 5 .

□ Proposition 1.22: Let A be an / x m matrix over a division ring D, B be an m x n matrix over D. Then rank A + rank B — m < rank AJ3 < min{rank A, rank B}. Proof: (i) For simplicity let C = AB, rA = rank A, r# = rank 5 , and rc = r a n k C There is an M G GLm(D) and TV € GLn(D) such that

M

B

/ j(r*)

= {

\

o)N-

Let C* = CN~X and A* = AM, then rankC* = rCj rank A* = rA, and A

*(^

o) = ^

Write A* = (AJ AJ), where A\ is / x r^ and A2 is / x (m — r ^ ) , then C* = (AJ 0). Since A* has r^ linearly independent columns, A\ has at least TA — {m — TB) linearly independent columns. Therefore rc > r^ + r^ — m. (ii) From C = AB we know that the rows of C are linear combinations of the rows of B. Therefore the row rank of C is less than or equal to the row rank of i?, i.e., rc < rs> Similarly, the columns of C are linear combinations of the columns of A. Hence rc < rAD Definition 1.11: Two m x n matrices A and B over a division ring D are said to be row equivalent if there is an m x m invertible matrix P such that A = PB. As in the case when D is a field we can prove that any m x n matrix over a division ring D is row equivalent to an echelon matrix, called its echelon normal form.

1.3. Matrix Representations

27

of Subspaces

Proposition 1.23: Let A be an m x n matrix over a division ring D, then A is row equivalent to a matrix of the following form: /0 •••0 1 * 0 ••• 0 0 0 0 ••• 0 0 0

* 0 * 0 1 * 0 0 0

0 0 0 0 0 0 0 0 0

* 0 * 0 1 *

* * *

0 * 0 * 0 *

* *

0 0 0 0 0 0

0 0 0 0 0 0

0 0

1 * 0 0

* 0

0 0 0

0 0 0

0

0 0

0

* o *

where * denotes some element of F. Further, let rank A = r, then all the elements below the r-th row of the above matrix are O's, the first nonzero element of each of the first r rows, from left to right, is 1, and the r l's belong to r different columns. Thus all the elements to the left of each of these l's in the same row are O's, all the other elements in the same column of each of them are also O's, and all the elements below and to the left of each of them are all O's, too. Hence if the first nonzero element 1 of the i-th row (1 < i < r) is located at column &;, then 1 < ki < &2 < * * * < K < n. □ We omit the proof the above proposition; interested readers may consult Wan 1992b, Theorem 5.27.

1.3

Matrix Representations of Subspaces

Now let P be an m-dimensional subspace of D^ and Vi, i^, • • •, vm be a basis of P. Then ^i, ^2, •••, vm are vectors in D^n\ We usually use the m x n matrix V2

\ vm /

28

Chapter 1. Linear Algebra over Division Rings

to represent the subspace P and write

(

Vl

\

\VmJ i.e., we use the same letter P to denote a matrix which represents the subspace P . We call the matrix P a matrix representation of the subspace P. It should be noted that a matrix representing an m-dimensional subspace is an 77i x n matrix of rank m. Of course, two m x n matrices P and Q both of rank m represent the same ra-dimensional subspace, if and only if there is an m x m invertible matrix A such that P = AQ. Elements of GLn(D) can be regarded as linear transformations of D^n\ Let T G GLn(D), then T carries (or transforms) the vector (xi,x2, • • • ,xn) of D^ into (a?i, x2, • • •, xn)T. That is, we have a map £> x GLn(D) —+ £>M ((xi, x2, • • •, z n ) , T) i—► (zi, x2, • • •, £ n )T. We also say that this is an action of GLn(D) on Z ) ^ , or GLn(D) acts on Z>(n). This action induces an action of GLn(D) on the set of m x n matrices over D and also on the set of m-dimensional subspaces of D^: if T G GLn(D) and P is an m x n matrix over D or an m-dimensional subspace of D^n\ then T carries P into P T . In fact, we have Proposition 1.24: If T G GLn(D) and P is an m x n matrix of rank m over /?, then P T is also an m x n matrix of rank m. Furthermore, if P and Q are both m x n matrices of rank m over Z), then there is an element T G GLn(D) such that P = QT. Proof: The first statement follows from Proposition 1.14. Now we prove the second statement. Let P and Q both be m x n matrices of rank m. By Proposition 1.15 there are (n — m) x n matrices B and C such that both

(5) - (?)

1.4. Systems of Linear

29

Equations

are n x n invertible matrices. Let

T = then T G GLn(D)

and P = QT.

(?)" ' ( * )

□

Corollary 1.25: The set of all non-zero vectors of D^ forms an orbit, i.e., a transitive set, under GLn(D). The zero vector (0,0, • • •, 0) is left fixed by every element of GLn(D). □ Proposition 1.26: The set of all m-dimensional subspaces of D^ an orbit under GLn(D). Proof: Use matrix representations of subspaces of D^n\ Proposition 1.27: The number of orbits of subspaces of D^

1.4

forms □

is n + 1. □

Systems of Linear Equations

Let D be a division ring, a tJ (i = 1,2, • • • , r a ; j = 1,2, • • • ,n) and 6t- (i = 1,2, • • • ,m) be elements of D, and xi, x2, • • •, £ n be n indeterminates over Z), then == bi&1 1 Gn^i G ll^i + + a a i122x#22 + + •* •' •' ++ cL CLlnXn lnxn a2iXi ++ «22^2 a22x2 H + • •• +h «o,22nnx^nn == bh 2 I «21^1 ,- ~x (1.8) o-miXi am2x2 H+ • hi ^m/i^n Q>mnXn == ^t m >J flml^l ++ «m2^2 is called a system of m linear equations in n indeterminates To use matrix notation, let A = (ay) l and W = < vr > . Then V = Vr-i+W, Vr-if)W — < 0 > , Vr-i + a is an (r — l)-flat contained in S, and W + a is a 1-flat (or line) contained in 5 . By induction hypothesis and the hypothesis of the lemma, A(Vr-i + a) is an (r — l)-flat contained in S' and A(W + a) is a 1-flat contained in S\ respectively. Therefore we can assume that A(Vr^

+a) = V;_, + a'

and A{ W + a) = W' + a\ where V^_x is an (r — l)-dimensional subspace of D^ and W is a 1dimensional subspace of Z?(n). Then V^x + a' C S' and W + a' C S'. By the definition of V K-i Q V

and

W C V.

We assert that

v' = vj!_1 + wi and y ; _ 1 n ^ , = < o > . From these two assertions it follows immediately that Vf is an r-dimensional subspace of D^. If K - i n ^ ^ < 0 > , then (!£_! + a') D (W7 + a') ^ {a 7 }. It follows that ( K - i + a) H (W + a) ^ {a} and then Vr_i fl W ^ < 0 >, which is a contradiction. Thus the second assertion is proved. Now let us prove the first assertion. We distinguish the following two cases. (a) D ± F 2 . Let v' be any element of V. Then v'+a' G S'. Since A(S) = S' and 5 = V + a, there is an element v EV such that A(v + a) = v' + a!. We may decompose v &s v = u + w, where u G Vr-i and w G W. Let X £ D but A ^ 0 , 1 . Then Au G V r _i. Denote the line joining the points Xu + a and v + a by /i, then /a = {xXu + (1 - x)v + a|x G £ } .

2.2. Fundamental Theorem of the Afhne

57

Geometry

Denote the line joining the points a and w + a by Z2, then /2 = {xw + a\x G D}. /1 and l2 intersect at a point which isA(A — 1 ) - 1 K ; + a. Thus l\ is the line passing through \u + a, A(A — 1) _ 1 K; + a, and v + a. Let ,4(Au + a) = u + a

and ,4(A(A - l) _ 1 u; + a) = w' + a,

where u' G V^_x and u/ G W . Then u' + a', u/ + a\ and u' + a' lies on the line which is the image of l\. Consequently, v' = \iv! + (1 — fi)wf for some fi e D. Since u' G K'-i

and

™' £ W>

we nave v

' £ K'-i + W'•

Therefore

v c y;.! + w. Conversely, for any u' G V ^ and w' G W7 let v' = uf + w'. Let A G D, but A 7^ 0,1. Then \u' G V ^ . Denote the line joining the points \u' + a' and v' + a' by l[, then l[ = {x\uf + (1 - z)vr be r linearly independent row vectors in Then r < n. Let V = < vi, v2, • • •, vr >, then V is an r-flat in D^n\ By Lemma 2.9 4 ( V ) is an r-flat in AGl(n,D). Since 0 G < ui, t>2, • * *, vr > and -4(0) = 0 , 4 ( V ) is an r-dimensional subspace of D^n\ We apply in­ duction on r to show that A(vi), Afa), - • -, A(vr) are linearly indepen­ dent. If r = 1, then v\ ^ 0. hence A(v\) ^ 0 and A(v\) is linearly independent. Assume that A(vi),A(v2), • • • , 4(tV-i) are linearly indepen­ dent. Let Vr-i — < vi, U2,- • •, vr-i > and W = < vr >. Then V = K - i + W and K_i fl W = < 0 > . By the proof of Lemma 2.9, A(V) = K'-i + W' and !//_! H W = < 0 >, where V / ^ = 4 ( K _ i ) and W = A{W). Clearly, ^ M W . - M K - i ) € K'-i and 4 ( v r ) G W. There­ fore 4(i>i), -4(^2)?'' * 5 4(iV-i), 4 ( v r ) are linearly independent. □ Now let us come to the proof of Theorem 2.7. Proof of T h e o r e m 2.7: The proof of the second statement is the same as the proof of Proposition 2.6(i) and hence is omitted. Now let us prove the first statement. We prove by induction on n. Consider first the case n = 2. Let A be a bijective map of AGl(n, D) which carries lines into lines. By the bijectivity of A, A carries parallel lines into parallel lines. Clearly, (0,0), (1,0), and (0,1) are three non-collinear points. By hypothesis, .4(0,0), ,4(1,0), and .4(0,1) are three non-collinear points, too. By Proposition 2.6(iii) there is an afEne transformation which carries .4(0,0), .4(1,0), and 4 ( 0 , 1 ) into (0,0), (1,0), and (0,1), respectively. Thus after subjecting A to this affine transformation, we can assume that 4 ( 0 , 0 ) = (0,0), 4 ( 1 , 0 ) = (1,0), and 4(0,1) = (0,1). We distinguish the following two cases. (a) D = F 2 . 4G(2,F 2 ) has only four points (0,0), (0,1), (1,0), (1,1). Therefore we must have .4(1,1) = (1,1). Thus A(x,y)

= (x,y) for all (x,y) €

AG(2,¥2).

60

Chapter 2. AfRne Geometry and Projective

Geometry

Hence Theorem 2.7 is proved for the case n = 2 and D = F2. (b) D ^ F 2 . Denote the line passing through (0,0) and (1,0) by /, and the line passing through (0,0) and (0,1) by /*. Then I = {(x,0)\x

e D]

and l* =

{{09x)\xeD}.

By hypothesis A(l) — I and A(l*) = /*. We can assume that A{x,0) = (x%0). We assert that a is an automorphism of D. Clearly a is a bijection and 0* = 0, la = 1. We have to prove that (s + t)ff = s° + ta and (sty = s't*

for all s,t e D.

These two formulas can be proved by the method of intersection and join, which was shown by the following figures.

Figure 1

Figure 2

2.2. Fundamental Theorem of the Afhne Geometry

61

The proof of the first formula reads as follows. Let (a, /?) be a point not on / and not on /*. Then a / 0 and (3^0. Passing through the point (0,0) draw the line lo =

{x(a,P)\xeD}.

Clearly, l0 ^ I and / 0 fl/ = {(0,0)}. (a, f3) is a point of the line l0 and distinct from (0,0). By Proposition 2.4(ii), passing through (a, /?) there is exactly one line parallel to /. Denote this line by /' , then l' = {(x,0) +

(a,/3)\xeD}.

There is a unique line passing through (s,0) and parallel to Z0, which is ls = {x(a,(3) +

(s,0)\xeD}.

The lines ls and V intersect at a point, denote it by P , then

P = (s + a,P). Denote the line joining (t, 0) and (a, 0) by lt, then lt = {x{t,0) + = {x(t-a,-P)

+

(l-x)(a,/3)\xeD} (a,/3)\xeD}.

Denote the line passing through P and parallel to lt by / s + t , then ls+t = {x(t - a, -P) + (s + a, P)\x e D}. The point of intersection of / and l9+t is (s + *,0), which is independent of the particular choice of the point (a, /?) ^ /, /'. In the above discussion if we replace (s,0) and (*, 0) by ( s a , 0) and (ta, 0), respectively, then (s +1,0) will be replaced by (sa + ta, 0). Therefore (s + t)a = sa + ta. Now let us prove the second formula. Since D ^ F 2 , there are at least two distinct lines passing through (0,0) and ^ /,/*. Let them be l0 =

{x{a,(3)\xeD}

and l'0 =

{x(a1J1)\xeD}.

62

Chapter 2. AfRne Geometry and Projective

Geometry

Then a, /?, a i , /?i are all ^ 0 and (a, /?) and (c*i, /?i) are linearly independent. Clearly, / o n / = { ( 0 , 0 ) } . Passing through (1,0), draw the line /1 = {x(a 1 ,^ 1 ) + ( l , 0 ) | a : G i ) } , which is distinct from / and not parallel to IQ. Denote the point of intersection of l0 and /i by Pi, then P1 = ((a - / ? / ? ! - % r ' a , (a - / ^ a x ) " 1 / ? ) . There is a unique line passing through («s,0) and parallel to /i, denote it by / s , then ls = {x{au/31) + (s,0)\xeD}. Denote the point of intersection of Z0 and ls by P-i, then P2 = (s(a - / ^ a i r V ^ a - ^ r 1 ^ ) " 1 / ? ) Denote the line passing through (t,0) and Pi by lt, then h = {x(t, 0) + (1 - *)((« - flST1^)-1**, (a - / J / J r V ) " 1 / ? ) ! * € £>} = {x(< _ ( a _ ftSr'ai)-1^ - ( a - / J / J r 1 ^ ) " 1 / ? ) +((a fifr-1^)-1^ (a - ^ r ' a i ) - 1 ) / ? ) ^ € Z?}. The line passing through P2 and parallel to /t is

/* = {*(* - (a - flST1**!)-1)^ - ( a - ^ r 1 ^ ) / ? ) +(s(a - ^ r

1

^)"

1

^ *(a - ^'1a1)'1P)\x

€ £>}.

The point of intersection of / and Zs* is (s£,0), which is independent of the particular choice of the lines IQ and lf0. If in the above discussion we replace (s,0) and (t, 0) by (s% 0) and (t a ,0) respectively, then (st, 0) will be replaced by (sata,0). Therefore (st)a = s°t°. Hence we have proved that a is an automorphism of D. After subjecting A to the bijective map of AGl(2, D) to itself (x,y)\—►

(x,y)a~\

which is of the form (2.4), we can assume that •4(3,0) = (3,0) for all

xeD

2.2. Fundamental Theorem of the Afhne Geometry

63

and 4 ( 0 , 1 ) = (0,1). By hypothesis the line /* = {(0,y)|y G D) passing through (0,0) and (0,1) is carried into itself by A. Therefore A(0,y) = (0,y T ) for all y G D, and we can prove in a similar way that r is an automorphism of D. The line passing through (s,0) and parallel to /* is {(0,y) + ( * , < % € ! > } , the line passing through (0, i) and parallel to I is {(s,0) + ( 0 , t ) | * € Z > } , and their point of intersection is (s,t). A(s,t)

Thus

= (s, T ) .

The line passing through (0,0) and (s,t) is {x(s,t)\x A(x(s,t))

= (xp(s,tT))

(2.5) G D}. Therefore

for alia; G D,

where /> is a bijection of Z). That is, A{xs,xt)

= (xps,xptT)

for all x e D.

(2.6)

On the other hand, substituting xs and xt for 5 and £, respectively, in (2.5), we obtain A(xs,xt) = (xs,xTtT) for alls G D. (2.7) Comparing (2.6) and (2.7), we obtain p = T = 1. Therefore ,4(s,t) = (s,t) for all s,t e D. Hence Theorem 2.7 is proved for the case n = 2.

Chapter 2. Affine Geometry and Projective

64

Geometry

Now let n > 3 and assume that Theorem 2.7 is true for the (n — 1)dimensional left affine space over D. We are going to prove that Theorem 2.7 is also true for the n-dimensional left affine space over D. Let 4 be a bijective map of AG ! (n, D) to itself which satisfies the conditions of Theorem 2.7. After subjecting A to the map (xu x2, • • •, xn) i—> A(xu x2, - - •, xn) - .4(0), where 0 = (0,0, • • •, 0), we can assume that 4 ( 0 ) = 0. Let A(ei) = (an, ai2, • - •, a t n ),

i = 1,2, • • •, n.

By Corollary 2.10 the n row vectors ( a n , a i 2 , • • • , ^ l n ) , («21> «22? • ' ' , «2n)? ' * * ? ( ^ n l , O n 2, * • • ,

ann)

are linearly independent. Let / an

a12

• • • aln \

«21

«22

'* '

#2n

\ «nl

«n2

* * '

Ann /

j. _

-* then T £ GLn(D).

I

I>

After subjecting 4 to the affine transformation (xux2,

• • •, xn) i—> (si, s 2 , • • •, ^ n ) T _ 1 ,

we can assume that 4(0) = 0 and 4(e z ) = et- for z = 1,2, • • •, n. Let A n _i = {(xi,x2,--'>xn-i,0)\xuX2,-'-,xn-i

e D},

then 4 n _ i is an (n — l)-flat containing the points 0, ei, e 2 , • • •, e n _i. By Lemma 2.9 4 ( A n _ i ) is an (n —l)-flat containing the points 0,ei,e 2 , • • • ,e n _i. By Proposition 2.3(iii) such a flat is unique. Therefore 4 ( A n _ i ) = 4 n _ i . Hence 4 induces a bijective map of 4 n - i to itself, which satisfies the condi­ tions of Theorem 2.7. By induction hypothesis, we have A(x1,---,xn-u0)

= {x1,---,xn_1,0y

IT

o\

l

l

Q*

J+(a1,---,aB_i,0),

65

2.2. Fundamental Theorem of the Affine Geometry

where a is an automorphism of D, T\ G GLn-i(D)\ and «i, • • • , a n - i £ £)• Since .4(0) = 0, we have ai = ••• = a n _i = 0. After subjecting A to a bijective map of the form (2.4)

(xi,x ( # 1 , 2#,-2,

• • ' •?, X£n)n )

1

►

(xi,x2,- • • i * n )

'Ti (

(P - 1 "

1

a-1

we can assume that A(xux2,,"

••• ,x ,^n-i,0) , z n _i,0) _ i , 0 ) = (xi,x (xi,x n _i,0) 2,-2 ,-- '•,x

and A(e n ) =: e„. As in the proof of case n = 2, we can prove that *4(zi,:r 2 , • • - , , £ „ ) = ( ^ i , « 2 , - " - ^ n )

for all (xux2,

• • • ,x n ) € A G ^ n , / ? ) ,

but the details will be omitted.

D

Remark 2.1: For the case D = F 2 and n > 3 if we assume only that the bijective map of AC?(n,F2) to itself carries lines into lines, then we cannot deduce that the map is of the form (2.4). In fact, when D = F 2 , every line of AG(n, F 2 ) contains only two points, and therefore any permutation of the points of AG(n,F2) carries lines into lines. Without any essential difficulty, Theorem 2.7 can be generalized as follows. Theorem 2.11: Let n and n' be integer > 2, and D and D' be division rings. Let A be a bijective map from the rc-dimensional left or right affine space An = AG\n,D) or AGr(n,D) to An, = AGl{n',D') or AGr(n\D'). Assume that A carries lines into lines. When n > 3 and D = F 2 , assume further that A carries planes into planes. Then n = n' and D is isomorphic to D' or anti-isomorphic to D'. In the first case, An and Ant are either both left affine spaces or both right affine spaces. If An = AGl(n,D) and Ani = AGl(n, D'), then A is of the form A(^i,x2,- • '

• ,xn)aT

? # n ) — V 3 '!? ^ 2 ? * *

+ (ai,a 2 ,« • • j ^ n )

Chapter 2. Affine Geometry and Projective

66

Geometry

for all (xu • • •, xn) G AGl{n, D); if An = AGr{n, D) and An, = AGr{n, £>'), then A is of the form A\xi,

x2, - • •, xn) = T[t(xux2l

• • •, xn)]a + *(); if A n = A) and A n , = AG^n, D'), then ^4 is of the form At(xi,x2,

• • •, xn) = (xi, x 2 , • • •, xn)TT + (a x , a 2 , • • •, a n )

for all *(a:i, x 2 , • • •, xn) G AGr{n, D)\ in the above two formulas r is an antiisomorphism from D to D', T G GLn(D'), and ai, a 2 , • • •, an G i?'. D

2.3

Projective Spaces and Projective Groups

Let n be an integer > 1 and Z)(n+1) be the (n + l)-dimensional row vector space over D. The 1-dimensional subspaces of Z)(n+1) will now be called points, the 2-dimensional, 3-dimensional, and n-dimensional subspaces of J}(n+1) w iU be called lines, planes, and hyperplanes, respectively. More gen­ erally, the (r + l)-dimensional subspaces of Z)(n+1) will be called projective r-flats, or simply r-flats (0 < r < n) in this and next sections. Thus 0-flats, 1-flats, 2-flats, and (n — l)-flats are points, lines, planes, and hyperplanes, respectively. An r-flat is said to be incident with an s-flat, if the r-flat as a subspace contains or is contained in the s-flat as a subspace. Then the set of points, i.e., the set of 1-dimensional subspaces of D^n+1\ together with the r-flats (0 < r < n) and the incidence relation among them defined above

2.3. Projective Spaces and Projective

Groups

67

is called the n-dimensional left projective space over D and is denoted by PGl(n,D). To be more concrete, we introduce the coordinate description of PGl(n, D). Let P be a point of PGl(n,D), that is, P is a 1-dimensional subspace of Z)( n+1 ). Let (x0, xi, • • •, xn) be a non-zero vector in P , then P =

{\(x0,xU'-,xn)\\eD}.

For any A € D*, we shall call the non-zero vector (Ax0, A#i, • • •, Xxn) a system of coordinates or simply the coordinates of the point P. We also say that (Arco, A#i, • • •, Xxn) is the point P. Clearly, a system of coordinates of a point P is uniquely determined up to a non-zero constant multiple of D from the left. According to the above definition of PGl(n^D)^ the set of r-flats of PGl(n,D) and the set of (r + l)-dimensional subspaces of Z)(n+1) are in one-to-one cor­ respondence. The r-flat F corresponding to an (r + l)-dimensional subspace U can be regarded as the set of points whose coordinates are the non-zero vectors of U. After adopting this point of view, we have Proposition 2.12: A hyperplane in PGl(n,D) is the set of points whose coordinates are non-zero solutions of a linear homogeneous equation in n +1 unknowns zo^o + x\a\ + • • • + xnan = 0,

(2.8)

where ao, ai, • • •, an 6 D are not all zero, and conversely. Proof: Let H be a hyperplane of PGl(n,D), then H corresponds uniquely to an n-dimensional subspace U of Z)(n+1) such that H is the set of 1dimensional subspaces contained in U. Let v i , v 2 , - - - , v n be a basis of £/", then (Vl\

\Vn

)

is a matrix representation of the n-dimensional subspace U. Consider the

68

Chapter 2. AfEne Geometry and Projective

Geometry

system of linear homogeneous equations in n + 1 unknowns

( Xx ° \ i

.

U\

=0.

\ Xn )

Since rankf/ = n, the solutions of this system of equations form a 1dimensional subspace P of tD^n+1\ Let (a 0 , ai, • • •, a n ) be a non-zero vector of P , then the n-dimensional subspace U is the solution space of the linear homogeneous equation in n + 1 unknowns (2.8). Hence the hyperplane H is the set of points whose coordinates are non-zero solutions of (2.8). Conversely, let (2.8) be given. Then the set of solutions of (2.8) form an ndimensional subspace U of D^n+1^, and hence the hyperplane corresponding to U is the set of points whose coordinates are non-zero solutions of (2.8). D

The linear homogeneous equation (2.8), whose non-zero solutions are all the points of a hyperplane if, is called the equation of the hyperplane H. Clearly, it is uniquely determined up to a non-zero constant multiple of D from the right. In the same way we can prove also Proposition 2.13: Any r-flat (0 < r < n) of PGl(n, D) is the set of points whose coordinates are non-zero solutions of a system of n — r independent linear homogeneous equations in n + 1 unknowns, and conversely. □ We define the dimension of an r-flat F in PGl(n,D) to be r and denote d i m F = r; however the corresponding (r + l)-dimensional subspace U is of dimension r +1. Moreover, we define the empty set (f> of points in PGl(n, D) to be of dimension —1 and denote dim an°^)> (ao , a i \ '" i a l^)> '' '•> (ao\

a

i \ '" i an^)

are

linearly independent.

□

According to Proposition 2.13 a line can be regarded as the set of points whose coordinates are non-zero solutions of n — 1 independent linear homo­ geneous equations in n + 1 variables. However as in the affine case a line in PGl(n,D) has also a parametric representation. Let P and Q be two dis­ tinct points on a line / with coordinates (x 0 , xi, • • •, xn) and (yo, yi, • • •, yn), respectively. Then / = {A(zo, £i, • • •, xn) + /i(t/o5 2/i, • • •, y n )|A, fi € D and are not both zero}. More generally, let P 0 , A , • * * ,Pr be r + 1 points not lying on any (r — l)-flat and let the coordinates of P t be {x0^\xi^\ • • •, x n ^ ) (0 < z < r). Then the unique r-flat passing through P 0 , Pi, • • •, Pr is {Ao(ar0(0), • ■ •, *„) + \i(x0ll),

• ■ •, «, (1 ») + • ■ • + A r (x„ ( r \ • • •, *„('>)},

where Ao, Ai, • • •, Ar € D and are not all zero. Now let us introduce the projective group. Any T £ GLn+i(D) the following way:

defines a point to point transformation of PGl{n, D) in

PG\n,D) — PG'(n,D) ( x 0 , x i , - - - , x n ) i—> (xo,Xi,---,x n )T.

^ ' '

This is well-defined. In fact, if (yo^yir " ^Vn) is another system of coor­ dinates of the point (#o,#i, • • • , x n ) , that is, there is a non-zero element XeD such that (y0,2/i, •••,»«) = A(s 0 ,3i, • • • ,ar n ), then (y 0 ,!/i, • • • ,y n )T = A(x 0 ,a;i, • • • ,xn)T. The transformation (2.9) is called a projective transfor­ mation of PGl(n, JD), and denoted by T. Denote the center of D by Z, then we have L e m m a 2.17: Two elements T and I \ of GLn+\(D) define the same pro­ jective transformation if and only if there is an A 6 Z* such that I \ = AT.

2.3. Projective Spaces and Projective

Groups

71

Proof: Suppose that there is an element \ e Z* such that ?\ = AT. Then under the projective transformation 2\, defined by 2\, we have (x 0 , xi, • • •, i n ) i—► (x 0 , xi, • • •, x„)Ti = (x 0 , Xi, • • ■, xn)\T = A(x 0 ,xi,--- ,x n )T. Hence T\ = T. Conversely, suppose that T\—T.

Let

i = (5u)oxxo)~1(fjlxxi). Thus A -1 xo _1 xiA = xo _ 1 xi. But xi can be any element of D, so A G Z*. □ In virtue of Lemma 2.17, it is natural to introduce the factor group PGLn+1(D)

=

GLn+1(D)/Zn+u

where

Zn+1={\lW\\€Z*}. We leave to the reader to prove that Zn+i is the center of GLn+\{D). The image of T G GLn+i (D) in PGLn+i (D) is also denoted by T and is a projec­ tive transformation. The group PGLn+1(D) is called the projective general linear group of degree n + 1 over D. Two geometric figures in the projective space are said to be projectively equivalent, if one of them can be carried into the other by a projective transformation. According to the Erlangen Program, the projective geometry

72

Chapter 2. Affine Geometry and Projective

Geometry

is the study of properties of geometric figures in projective space which are invariant under the projective general linear group. Clearly, r-flats are carried into r-flats by any projective transformation. Thus the property for a geometric figure being a flat and the dimension of a flat are invariant under the projective general linear group. Now let us study the transitivity properties of

PGLn+i(D).

Proposition 2.18: (i) The group PGLn+i(D) PG (n,D).

is doubly transitive on the set of points of

(ii) The group PGLn+i(D) of points of PG\n,D).

is transitive on the set of independent triples

l

(iii) The group PGLn+i (D) is transitive on the set of independent sets of r + 1 points (0 < r < n). (iv) The group PGLn+i(D)

is transitive on the set of r-flats (0 < r < n).

(v) The group PGLn+i (D) is transitive on the set of (n + 2)-subsets of points such that any n + 1 of them are independent. Proof: It is sufficient to prove (iii) and (v), since (i) and (ii) are special cases of (iii), and (iv) is a consequence of (iii) by Proposition 2.15. Let P 0 , P i , ' ' • ,Pr be r + 1 independent points and the coordinates of P t be (a l0 , flti, • • •, din) (0 < i < r). It is easy to see that the following r + 1 points ^

= (l,0,--- > 0), J B 1 = ( 0 , l , - - - , 0 ) , - - - , £ r = ( 0 , 0 , . . . , 0 , l , 0 , - - . , 0 ) ,

where Ei (0 < i < r) is the point whose i-th coordinate is 1 and all the other oordinates are 0's, are independent. If we can find a projective transforma­ tion which carries P 0 , Pi, • • •, P r into J?0, i?i, • • •, Er, respectively, then (iii) will follow. Form the (r + 1) x (n + 1) matrix /

CLQQ aoi

•••

a0n

a\o an

"'

a\n

\ aro ari

\

" - arn )

2.3. Projective Spaces and Projective

Groups

73

It follows from Lemma 2.16 that the above matrix is of rank r + 1. We can supplement n — r more rows to the above matrix such that the obtained matrix

rp _

/

aoo

GOI

* * *

CLOn

I

a

r0

#r0

* * '

arn

\

a>nO

an2

-• •

ann

\

)

is invertible. Then (a z0 , aiu • • •, a^T'1 = et-, i = 0,1, • • •, r, and hence PiT = E{, i — 0,1, • • •, r. This proves (iii). Now let Po> Pi,' •' j Pn+\ be n + 2 points such that any n + 1 of them are independent. It is clear that any n + 1 of the following n + 2 points ^ , = (l,0,---,0),^1 = (0,l,0,.-.,0),--.,^n = (0,0,.-;,0,l), £ = (1,1,...,1) are independent. If we can find a projective transformation which carries P(b A r " ) ^n+i into E0, J5I, • • •, i? n , J5, respectively, then (v) will follow. By (iii), there is a projective transformation T such that PiT = Ei,i = 0,1, • • •, n. Assume that Pn+iT = (x 0 , #i? • • •, xn). Then any n + 1 of the n + 2 points E0, E1, • • •, 2?n, and (x 0 , #i, • • •, xn) are independent. It follows from Lemma 2.16 that x0 ^ 0,^1 ^ 0, • • • , x n 7^ 0. Clearly the projective transformation / xo-1

\

\"

Xn-1 I

leaves each of the n + 1 points 2?0, # i , • • ■, En invariant and carries the point (x 0 , #i, • • •, xn) into £*. This proves (v).

D

74

Chapter 2. Affine Geometry and Projective

Geometry

The coordinates (xo, #i, • • •, xn) of a point P £ PGl(n, D) introduced at the beginning of this section will now be called the homogeneous coordinates of the point P. When x 0 ^ 0, let & = x 0 - 1 x t ( i = 1,2,---jn), then we can choose (l,£i, • • • ,£ n ) to be the homogeneous coordinates of P and call (£i? *' * >£n) the non-homogeneous coordinates of P . Notice that, only those points (x 0 , Xi, • • • , x n ) for which Xo ^ 0 have non-homogeneous coordinates and they will be called finite points. Points with homogeneous coordinates (0, Xi, • • •, x n ) are called points at infinity. The set of points at infinity is a hyperplane in PGl(n,D), whose equation is x0 = 0

and this hyperplane is called the hyperplane at infinity. The finite points of PGl(n,D) are in one-to-one correspondence with the points of AGl(n,D) ( i , 6 , - - - , 6 i ) — > (6,•••>£»)• Consider the projective transformation (x 0 , xi, • • •, x n ) i—► (y0, yw-y

Vn) = (a*, xu • • •, x n )T,

(2.10)

where ^ — (^«i)ofa b> \c

dj

| €

GL2(D).

In non-homogeneous coordinates, (2.15) can be expressed as (a ++ sxc)~ xd). y == (a c ) " l1(b ^ ++ xd), y--

(2.16)

w

X = where yt/ == y^yi = aXQ^ Z i - Let 2/0 V and x

faa b^ bY1I

\c x c dd}j1

- 1((a'a' V b'\

" I

~ \[c' c' d'd'j-

Then \ _ _fa> ( a' bb'' \\ fa ( a 6b\ \ _ fl / 1 00\\ faa bb\\ ( fa'a' d VV\_ \c c d)\d d) U ')~ d' ) ~"U \c' d' ){c d ) ~ \0 *) [c dj--{0 l1 j) -'

(2.17)

Using the above identity, it can be readily verified that (a + (a xc)(-b'-&' + ax) -== (b(b++xd)(d xd)(d'f --- c'x). + ax) + xc)(Then we also have 1 1 (-&' + + a'x){d' a'x)(d' -- c'x). . yy = (-V c'x)-

(2.18)

Definition 2.1: Let X\,x2,x3 and x4 be four points on the projective line PGl(\,D) in non-homogeneous coordinates. They are said to form a har­ monic set, if (x11 --- x£44) == —1. -1. - x3)(x1 ---xx3)3)~11(x ) 11(x (x22 -(x2 --- x4)~

(2.19)

□ l

We want to characterize those bijective maps of PG (l, D) which carry har­ monic sets into harmonic sets. At first we give the following definition.

82

Chapter 2. AfRne Geometry and Projective

Geometry

Definition 2.2: Let D and D' be division rings, a bijective map cr : a —> aa(a € D) from D to D' is called semi-isomorphism from D to £)' if for any a,b E D the following conditions hold: {a-rbY

=aa + b%

(2.20)

(a&a)* = a'b'a*,

(2.21)

and r = 1. When D — D', it is called a semi-automorphism

(2.22) of D.

□

Set b = 1 in (2.21) and (2.22), we deduce (a2)* = (a*) 2 .

(2.23)

We have the following beautiful theorem of Hua. Theorem 2.25 (Hua): Every semi-isomorphism from a division ring to another division ring is either an isomorphism or an anti-isomorphism. In particular, every semi-automorphism of a division ring is either an automor­ phism or an anti-automorphism. Proof: Let D and D' be division rings and a : a —» a°(a £ D) be a semiisomorphism from D to D'. For any a,b,c £ D, from (2.20) and (2.21) we deduce (abc + cba)a = ((a + c)b(a + c) - aba - cbcf = (a* + ca)ba(a° + ca) - aabaaa - cabac° = aabaca +cab°a°.

(2.24)

Then from (2.20)-(2.24) we deduce {(abY - a*b") ((aby - b°a°) = ((abY)2 + a°(V)2a* - (a°V(aby + (ab)aWa') = ((ab)2)" + (ab2af - (ab(ab) + (ab)ba)' = 0. Since D' is a division ring, we have (ab)a = aaba or baaa for all a, b £ D. If D' is a field, then, clearly, Theorem 2.25 is true. Now let D' be not a

83

2.5. One-dimensional Projective Geometry

field, i.e., the multiplication of D' is not commutative. We distinguish the following two cases: (a)

There is a pair of elements a, b G D' such that (abY = aaba ±

¥aa.

We prove by the method of "infection" that for any c,d G D (cdy

=cad%

i.e., cr is an isomorphism from D to D'. At first we assert that (cb)a = caba and (acy = a° ca. Suppose that there is a c G D such that (cb)a = baca ^

i.e., (x + yr = i ( ( 2a; 2 xrr - ( 22yy)r' ) .. (* + »)* = 2(( Let y = 0 in the above equation, then we have a

X x f f :=

-(2xY.

; = 25'V 2l»"'

Therefore a a (* (x + y)a = y)x" =x+a+y y°-.

Let X\ = x,x2 = 1 — x, and £3 = 2x\x2^ then Xi,£2?#3? and 0 form a harmonic set. Therefore x^x2,xl, and 0 form a harmonic set, i.e., (x [x2)

[x22 —xx3)(x [x 1l 3)[x

—* * x3 /3)

X

l

x 1~~ = —1. *■'

2.6.

Comments

87

It follows that £3 = 2x\xa2. Hence (2x(l-x)Y

2xa(l-x)%

=

from which follows

(*2r = K) 2 . Thus

(xy + yxY = ((x + y)2 - x2 - y2f = (** + y * ) 2 - ( * * ) 2 - ( y * ) 2 = x*ya + yax*.

From 2xyx = x(yx + xy) + (yx + xy)x - (x2y + yx2), we deduce {xyxY =

x°y°x°.

Therefore a is a semi-automorphism of D.

2.6

D

Comments

The presentation of the affine space and projective space over any division ring in Section 2.1 and 2.3, respectively, are standard, and we follow mainly Wan 1993. Section 2.2 is adopted from Hua 1951, but with modifications and correc­ tions. The fundamental theorem of the affine geometry over the field of real numbers R can be found in Veblen and Whitehead 1933 and that over any division ring D appeared in Hua 1951. But for the case of dimension n > 3 and D = ¥2 the hypothesis of the theorem should be modified, which was pointed out in Wan 1961. The fundamental theorem of the projective geometry over any division ring is a rather old result and can be found in many books, e.g., Baer 1952 and Artin 1957. In Section 2.4 we derive it from the fundamental theorem of the affine geometry over any division ring, which is proved in Section 2.2. If one likes, one can also prove the fundamental theorem of the projective geometry over any division ring directly and then deduce the fundamental theorem of the affine geometry over any division ring from it.

88

Chapter 2. Afiine Geometry and Projective

Geometry

All the beautiful results and elegant proofs in Section 2.5 are due to L.K.Hua, for Theorem 2.25, cf. Hua 1949 and for Theorem 2.26 and 2.27, cf. Hua 1951.

Chapter 3 Geometry of Rectangular Matrices 3.1

The Space of Rectangular Matrices

Throughout this chapter let D be a division ring and m, n be integers > 2. Denote the set o f m x n matrices over D by M.mXn(D). When m = n, we write simply Mn(D) for Mnxn(D). MmXn(D) is the space we are going to study in this chapter and we call it the space ofmxn matrices over D, or simply the space of rectangular matrices if the size of the matrices is clear from the context. We also call the matrices in MmXn(D) points of the space. With the space M.mxn(D) we associate naturally a group of motions which consists of transformations of the form X H—> PXQ + R

for all X G Mmxn{D),

(3.1)

where P G GLm(D), Q G GLn{D), and R G Mmxn(D). Clearly, (3.1) is bijective. Denote this group by GmXn(D). We begin with the study of the transitivity properties of the group GmXn{D) acting on the space Mmxn{D). Proposition 3.1: The group GmXn(D)

acts transitively on

MmXn{D)-

Proof: Let X\ and X2 be any two m x n matrices over D. transformation X i—► X + (X2 — X\) carries X\ to X2.

Then the a

Definition 3.1: Let Xi, X2 G Mmxn(D). They are said to be of arithmetic distance r, denoted by ad(Xi, X2) = r, if rank(Xi — X2) = r. When r = 1, 89

90

Chapter 3. Geometry of Rectangular

then they are said to be adjacent (or coherent).

Matrices □

The arithmetic distance fulfills the three requirements for the distance func­ tion in a metric space. Proposition 3.2: Let Xu X2, X3 G Mmxn(D).

Then

1° ad(Xi, X2) > 0; ad(Xu X2) = 0 if and only if X1 = X2. 2° a d ( X i , X 2 ) = a d ( X 2 , X i ) . 3° ad(X l 5 X 2 ) + ad(X 2 , X 3 ) > ad(Xi, X 3 ). Proof: 1° and 2° are clear, and 3° follows from Proposition 1.21.

□

Proposition 3.3: The elements of the group GmXn(D) leave the arithmetic distance between any pair of points of M.mxn{D) invariant. Moreover, for any r with 1 < r < min{m, n } , the set of pairs of m x n matrices over D of arithmetic distance r forms an orbit under Gmxn(D). Proof: The first statement is clear. Let us prove the second statement. Let Xi, X2 be a pair o f m x n matrices over D which are of arithmetic distance r, i.e., rank(Xi — X2) = r. Let R be an m x n matrix of rank r. It is enough to show that there is an element of Gmxn(D) which carries X\ and X2 to 0 and R, respectively. Clearly, the transformation X i—> X — X\ carries X\ to 0 and X2 to X2 — X\. We have rank(X 2 — Xi) = r. Since R is also of rank r, by Corollary 1.19 there is an element P G GLm(D) and an element Q G GLn(D) such that P(X2 — X\)Q = R. Clearly, the transformation X i—► PXQ leaves 0 invariant and carries X2 — X\ to R. D Therefore the arithmetic distance between any pair of points of MmXn(D) is a geometric invariant under the group Gmxn(D), so is, in particular, the adjacency of a pair of points of M.mxn{D). We would like to characterize the elements of the group GmXn(D) by as few geometric invariants as pos­ sible. We will see that the invariance of the adjacency of pairs o f m x n matrices alone is sufficient to characterize the transformations of the form (3.1) to within automorphisms of D if m ^ n. More precisely, we have the following fundamental theorem of the geometry of rectangular matrices over any division ring.

3.1.

The Space of Rectangular

91

Matrices

Theorem 3.4: Let D be any division ring, m and n be integers > 2, and A be a bijective map from Mmxn(D) to itself. Assume that both A and A'1 preserve the adjacency of pairs of m x n matrices, i.e., for any X i , X2 G Mmxn(D), X\ and X2 are adjacent if and only if A(Xi) and A(X2) are adjacent. Then when m ^ n, A is of the form ,4(X) = PJSTQ + R

for all X G Mmxn{D),

(3.2)

where P G GLm{D), Q G GLn(D), ReM mx.n{D), cr is an automorphism of D, and X a denotes the matrix obtained from X by applying a to all the entries of X. When m = n, in addition to (3.2), we have also A(X)

= P\XT)Q

+R

for all X G MmXn(D),

(3.3)

where r is an anti-automorphism of D. Conversely, any map of Mmxn(D) to itself of the form (3.2) (and of the form (3.3), when m = n) is bijective, and both the map and its inverse map preserve the adjacency of pairs of m x n matrices. After some preparations in Sections 3.1 - 3.3, the proof of Theorem 3.4 will be given in Section 3.4. Definition 3.2: Let X, X' G Mmxn(D). When X ^ X', they are said to be of distance r, denoted by d(A", X') = r, if r is the least positive integer for which there is a sequence of r + 1 points Xo, «X"i, • • •, Xr with XQ = X and Xr = X' such that X{ and Xi+i are adjacent, z = 0, 1, • • • , r — 1. When X = X', we define d(X, X) = 0. □ Proposition 3.5: For any two points X, X' G A^ m X n(^)j ad(X, X') = d(X,

X').

Proof: When X = X', clearly we have ad(X, X ) = d(X, X) = 0. Now let X ^ X'. Assume that ad(X, X') = r, i.e., rank(X - X') = r. By Proposition 1.17 there is an element P G GLm(D) and an element Q G GLn(D) such that P(X-x')3=(/(r)

0(m_r,n_r)).

Chapter 3. Geometry of Rectangular

92

Matrices

Let Hi = P-1 ( 7

%

0 ( r o - t -. n - 0

) Q~\

i = 1, 2, • • ■, r.

Then Xo

=

-X"? -Xi = X — i?i, X2 = A. — it25 * * * > XT = X — Rr = X

is a sequence of r + 1 points of MmXn(D) such that X{ and X,+i are adjacent, * = 0, 1, 2, • • •, r - 1. Therefore d(X, X') < r = ad(X, X ' ) . Conversely, suppose that d(X, X') = r'. Then there is a sequence of r' + 1 points X 0 , Xi, X2, • • •, Xrt with Xo = X and J£r/ = X ' such that X{ and X t + i are adjacent, i = 0, 1, 2, • • •, rf — 1. By Proposition 1.21 d(X, X') = r' = r £ * rank(Xt- -

Xi+1)

t'=0

> rank(X 0 - Xr.) = rank(X - X') = ad(X, X ' ) . Therefore ad(X, X') = d(X, X'). D

Corollary 3.6: Let A be a bijective map from MmXn(D) to itself and as­ sume that both A and A'1 preserve the adjacency of pairs oimxn matrices, then A also preserves the arithmetic distance between any pair oimxn ma­ trices, i.e., for any X, X' G Mmxn(D), ad(X, X') = *d(A(X), A{X')). Proof: Suppose that A preserves the adjacency of pairs oimxn i.e., for any Xi, X2 G Mmxn(D), adjacency of A(Xi)

and A(X2).

matrices,

the adjacency of X\ and X2 implies the Then for any X, X' G

d(X, X') > d{A{X),

A{X')).

By Proposition 3.5 we have ad(X, X') > ad(A(X),

A{X')).

Similarly, if A~x preserves also the adjacency, then a d ( ^ ( X ) , A{X'))

> ad(X, X').

MmXn(D),

93

3.2. Maximal Sets of Rank 1

Therefore, if both A and A 1 preserve the adjacency, then for any X, X' € Mmxn(D), ad(X, X') = ad(A(X), A(X')).

□

3.2

Maximal Sets of Rank 1

Definition 3.3: Let M. be a non-empty set of points in the space M.mXn(D). M. is said to be a maximal set of rank 1, if any two points of M. are adjacent and there is no other point outside M., which is adjacent to each point of M. D Proposition 3.7: A maximal set of rank 1 in AimXn{D) is carried into a maximal set of rank 1 under any transformation of the form (3.1). Proof: It follows from Proposition 3.3.

□

Proposition 3.8: Both f/Zll

Mi = I

#12

0

0

E n , ^12, • • • , xln

Vo and

( J/u

M[ =

0

2/21

e

D

0 /

o 0 0

0\ 0 S/ii, 2/2i, • • • , 2/mi € D

\J/ml

(3.4)

0

(3.5)

0)

are maximal sets of rank 1. Moreover, every maximal set of rank 1 can be carried under a transformation of the form (3.1) to either Ai\ or M.[. Proof: Clearly, any two points of M\ (or M[) are adjacent. That there is no other point outside .Mi (or Ai[) which is adjacent to every point of M.\ (or M!x, respectively) will follow from the argument of next paragraph. Let M. be a maximal set of rank 1 contained in A4mXn(D). Clearly M. cannot consist of a single point. Let X and X' be a pair of points in A4, then they are adjacent. By Proposition 3.3 there is a transformation of

94

Chapter 3. Geometry of Rectangular

Matrices

the form (3.1) which carries X and X' into 0 and E n , respectively. The transformation will carry M into a maximal set of rank 1 containing 0 and E\\. Thus we may assume that M contains 0 and En. Let X\ be another point of M. Since X\ and 0 are adjacent, X\ is of rank 1. Therefore we may write ( Gl&l

a2bi *i

CL2b2

0-2K

=

\ambi

amb2

where ai, a2, • • •, am 6 D and are not all zero, and &i, &2> • • •, bn € D and are not all zero. Since X\ and En are also adjacent, X\ — En is of rank 1. Therefore aibj = 0, i = 2, 3, • • •, m; j — 2, 3, • • •, n. If 62, 63 thus

, bn are not all zero, then a2 = a 3 = • • • = a m = 0, a\ ^ 0, and (d\bi

0 *i

aib2 0

0

=

V 0

0

(3.6)

0 y

Otherwise, we have

*i

where b\ ^ 0 and a2> a 3 , and

/ ai&i a2bi

0

\a>mb\ 0

0/

=

(3.7)

a m are not all zero. The intersection of M\

JM'X

{xEu

\xeD}

(3.8)

is evidently not a maximal set. If M contains a point outside (3.8), say a point of the form (3.6) with ai ^ 0 and b2, 63, • • •, bn not all zero, then none of the points of M can be of the form (3.7) with &i ^ 0 and a2> «3, • • •, am not all zero. Similarly, if M contains a point of the form (3.7) with h ^ 0 and a2, «3, • • •, «m not all zero, then none of the points of M can be of the form (3.6) with «i ^ 0 and 62, b3, • • •, bn not all zero. Clearly every pair of points in Mi (or M[) are adjacent. Therefore M is either Mi or M J . D

95

3.2. Maximal Sets of Rank 1 Corollary 3.9: A maximal set of rank 1 is either of the form 0

0

\ 0

0

Q + R xn, a;i2, • • •, Zi„ G D

(3.9)

0 /

or of the form / 2/ii 2/21

0 0

where P G GLm(D),

0\ 0

(3.10)

Q + R 2/n, 2/21, ••*, 2/mi € D

Q G GLn(D),

and # G A^ m X n(^).

D

The proof of Proposition 3.8 gives the following corollaries. Corollary 3.10: Given any pair of adjacent points in Mmxn(D), there are two and only two maximal sets of rank 1 containing both of them. □ Corollary 3.11: The intersection of two distinct maximal sets of rank 1 which contains more than one point in common can be carried into (3.8) under a transformation of the form (3.1). □ Definition 3.4: If the intersection of two distinct maximal sets of rank 1 contains more than one point in common, then it is called a line. □ Corollary 3.12: There is one and only one line passing through any pair of adjacent points. □ Corollary 3.13: The parametric equation of a line in AdmXn

is

{lpxq + R\xeD}, where p is a nonzero m-dimensional row vector over D, q is a nonzero ndimensional row vector over D, and R G MmXn(D). The parametric equa­ tion of a line in the maximal set of rank 1 (3.4) is (

/an 0

ai2 0

V o

o

Q>ln\

0

x 0 /

/ &11

+

&12

0

0

0

V o

o

0 /

xeD

Chapter 3. Geometry of Rectangular

96

Matrices

where (an, #12, • • •, Q>\n) 7^ 0, and the parametric equation of a line in the maximal set of rank 1 (3.5) is f / an

0

«21

0

^

0

K\aml

•••

&21

0

Uml

0

x+

0/

x £D 0/

where ' ( a n , a 2 i, • • •, a m l ) ^ 0.

D

Furthermore, we have Proposition 3.14: Two maximal sets of rank 1 which have only one point in common can be carried simultaneously under the group Gmxn{D) to either (3.4) and

i I 0

0 \

0

Xix

Xi2

Z2n

0

0

0

I\ 0

0

o /

#21, #22, '" , X2n € D

(3.11)

2/12, 2/22, • • • , 2/m2 €

(3.12)

or to (3.5) and /0

J/12

0

0

y22

0

V 0 ym2

0\ 0 D

0

Proof: Let A4 and M! be two maximal sets of rank 1 which have only one point in common. By Propositions 3.1 and 3.8 we can assume that the common point of M and M! has been carried to 0 and M has been carried into (3.4) or (3.5) by an element of G m x n ( J D). Consider the case when M has been carried into (3.4). (The other case can be treated in a similar way.) Let M! be carried into M" and let

X2 =

«21

V «ml

«12

«ln \

«22

Q>2n

3.

99

3.3. Maximal Sets of Rank 2 Since M 0 M' = { 0 }, we have A = 0. Let / #11

X12

#ln

^21

^22

Z2n

\ Xmi

Xm2

X =

\

be any point in M. Then ad(X, 0) = 2, i.e, rank X = 2. Denote X," =

^X t 'i, Xi2)

? '^inj'i

1

=

I?

^)

ra.

Since X is of arithmetic distance < 2 from every point of M. and not of arithmetic distance 2 from every point of M, we have (X2\ rank

= 1.

Vx m / Similarly, rank

= 1.

If there is some i (3 < i < m) such that xt- ^ 0, then there exists Ik £ D (k = 1, 2, • • •, m) such that ra.

#A; — 'fc^n ' & — -•-> A

It follows that r a n k X = 1, which is a contradiction. Therefore X{ = 0 for i = 3, 4, • • •, ra, and X is a matrix of rank 2 of the form

^21

#22

%2n

0

0

0

V 0

0

o )

All matrices of rank 2 and of the above form constitute M.

100

Chapter 3. Geometry of Rectangular

Let 2/12

2/m\

2/21

2/22

2/2n

\ 2/ml

2/m2

(2/n

Y =

* '*

Matrices

2/mn /

be a point which is of arithmetic distance 1 from 0, of arithmetic distance < 2 from every point of Af, and not of arithmetic distance 2 from every point of Af. Then r a n k F = 1. Denote (Vi = 2/;i, 2/i2, " • ' , 2/m),

i = 1, 2, • • •, m.

If y3 zfi 0, then there exists Ik G D (k = 1, 2, 4, 5, • • •, m) such that Vk = 42/3, fc = 1, 2, 4, 5, •••, m. The transformation /l

\

-h ~/ 2 1 -U

1 X

X

l

carries A^i, M.2, and AT into themselves, and carries Y to 0

y* =

2/3

0

Vo/ Then Y* is of arithmetic distance > 2 from every point of Af, which is a contradiction. Therefore y3 = 0. Similarly y4 = y5 = • • • = ym = 0. Hence F is a matrix of rank 1 of the form 2/12

fyn

2/21

2/22

0

0

\ 0

0

•••

2/in\

2/2n

0

3.3.

Maximal

Sets of Rank

101

2

Therefore C is (3.14). C o r o l l a r y 3 . 1 7 : A m a x i m a l set of rank 2 is either of t h e form #12

•'•

#21

#22

''*

X2n

0

0

•••

0

V0

0

•••

0 /

IXX\

Xln\

Q+R

# i i , #12, • • *, xin, ^ 2 1 , £22, • • • , x2n

G i?

or of t h e form f

[ Vu T/21

I

\yml

2/12

0

•••

0\

2/22

0

•••

0

Vm2

0

•••

where P G GLn(D),

g + i? 2/ll, 2/21, • ' * , 2/ml, 2/12, 2/22, * * * , 2/m2 G £>

0)

Q G GLn{D),

a n d i? G M m x n ( 5 ) .

°

Now let us s t u d y t h e intersection of a m a x i m a l set of rank 1 and a m a x i m a l set of rank 2, when it is nonempty. We can assume t h a t their intersection contains 0. By Corollary 3.9, a m a x i m a l set of rank 1 containing 0 is either of t h e form /Xn

#12

0

A°° \ 0

l

0

Q

£ n , ^12, • • *, xln

G D

0 /

or of t h e form 2/21 \ymi where P € GLm(D),

0

0\ 0

0

0)

Q G GLn(D).

Q

yn, 2/21, • • • , ym\ € D

By Corollary 3.17 a m a x i m a l set of rank

2 containing 0 is either of t h e form Xl„\ X21

X22

X2>

0

0

0

I Vo o

o )

Q

X\l,

X12, • • • ,. X i „ , 1 2 1 , 222, • • • , X2n € £>

102

Chapter 3. Geometry of Rectangular

Matrices

or of the form f

[

/ 2/ii

2/12

0

•••

0\

2/21

2/22

0

•••

0

KVml

2/m2

0

•■■

Oj

Q

2/11? 2/21, • • • , 2/ml, 2/12, 2/22, * * • , 2/m2 €

£

where P G GLm(D) and Q G GLn(D). Therefore we can assume that the maximal set of rank 1 containing 0 and the maximal set of rank 2 containing 0 are carried under a transformation of the form (3.1) into one of the following four cases: (a)

r / yn p

f/xn 0

X12 ••• 0 •••

xln\ 0

IV o

o

0 /

yi2

•••

2/21

2/22

• ••

2/2n

0

o

•••

0

\ 0

0

Xu,

X12, ■•• , Xin €

D

yin\

Q

2/11, 2/12, • • • , 2/ln, 2/21, 2/22, • • • , 2/2n € D

o /

GO f /Zll

0

Zl2

• ••

Xln\

0

•••

0 £ i i , #12, •• *, xin € £>

IV o

0 /

o

/ 2/n

2/12

0

• • •

0 \

2/21

2/22

0

•••

0

2/11, 2/21, • * • , 2/ml, 2/12, 2/22, ' ' ' , 2/m2 £ D

I

\2/ml

2/m2

0

•••

0/

(c)

/an

0

•••

0\

X21

0

•••

0 E n , #2i, • • • , x m i G D > ,

I V Zml

0

0/

103

3.3. Maximal Sets of Rank 2 f

/ 2/n

3/12

•

2/ln

2/21

J/22

■••

2/2n

0

0

•■•

0

[ \ 0

0

P

Q

2/11, 2/12, • • • , 2/ln, 2/21, J/22, • • • , 2/2n € I>

o/

(d) f / aril

0

a;2i

0

0\ 0 z n , ^2i, • • • , ^mi € D

l V zmi f

I

0

0/

/ 2/n

2/12

0

•••

0 \

2/21

2/22

0

•••

0

V2/ml

J/m2

0

where P € GLm(D)

••

Q | J/ll,

2/21, • • • , Vml, 2/12, 2/22, • • • , 2/m2 G -D

0/

and Q €

GLn{D).

Consider first Case (a). Clearly we can assume that Q = l(n\

Let

* — \PiJ h12

^22

0 0

Zln\ Z2n

P21

P22 Xu

X12

Xi n

.#21

X22

X2. n J

0