The optimal version of Hua's fundamental theorem of geometry of rectangular matrices

86 Pages·2014·0.568 MB·English

by  Peter Semrl

MEMOIRS
of the
American Mathematical Society
Volume 232 • Number 1089 (first of 6 numbers) • November 2014
The Optimal Version of Hua’s
Fundamental Theorem of Geometry
of Rectangular Matrices
ˇ
Peter Semrl
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 232 • Number 1089 (first of 6 numbers) • November 2014
The Optimal Version of Hua’s
Fundamental Theorem of Geometry
of Rectangular Matrices
ˇ
Peter Semrl
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Sˇemrl,Peter,1962-
TheoptimalversionofHua’sfundamentaltheoremofgeometryofrectangularmatrices/Peter
Sˇemrl.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 232,
number1089)
Includesbibliographicalreferences.
ISBN978-0-8218-9845-1(alk. paper)
1.Matrices. 2.Geometry,Algebraic. I.Title.
QA188.S45 2014
512.9(cid:2)434–dc23 2014024653
DOI:http://dx.doi.org/10.1090/memo/1089
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Contents
Chapter 1. Introduction 1
Chapter 2. Notation and basic definitions 5
Chapter 3. Examples 9
Chapter 4. Statement of main results 27
Chapter 5. Proofs 29
5.1. Preliminary results 29
5.2. Splitting the proof of main results into subcases 50
5.3. Square case 55
5.4. Degenerate case 58
5.5. Non-square case 64
5.6. Proofs of corollaries 67
Acknowledgments 71
Bibliography 73
iii

Abstract
Hua’sfundamentaltheoremofgeometryofmatricesdescribesthegeneralform
of bijective maps on the space of all m×n matrices over a division ring D which
preserve adjacency in both directions. Motivated by several applications we study
a long standing open problem of possible improvements. There are three natural
questions. Canwereplacetheassumptionofpreservingadjacencyinbothdirections
by the weaker assumption of preserving adjacency in one direction only and still
get the same conclusion? Can we relax the bijectivity assumption? Can we obtain
an analogous result for maps acting between the spaces of rectangular matrices of
different sizes? A division ring is said to be EAS if it is not isomorphic to any
proper subring. For matrices over EAS division rings we solve all three problems
simultaneously, thus obtaining the optimal version of Hua’s theorem. In the case
ofgeneral division rings weget such anoptimal result only for square matricesand
give examples showing that it cannot be extended to the non-square case.
ReceivedbytheeditorMay28,2012,and,inrevisedform,December4,2012.
ArticleelectronicallypublishedonFebruary19,2014.
DOI:http://dx.doi.org/10.1090/memo/1089
2010 MathematicsSubjectClassification. Primary15A03,51A50.
Keywordsandphrases. Rank,adjacencypreservingmap,matrixoveradivisionring,geom-
etryofmatrices.
TheauthorwassupportedbyagrantfromARRS,Slovenia.
Affiliationattimeofpublication: FacultyofMathematicsandPhysics,UniversityofLjubl-
jana,Jadranska19,SI-1000Ljubljana,Slovenia,email: [email protected].
(cid:3)c2014 American Mathematical Society
v

CHAPTER 1
Introduction
Let D be a division ring and m,n positive integers. By Mm×n(D) we denote
the set of all m×n matrices over D. If m = n we write Mn(D) = Mn×n(D). For
an arbitrary pair A,B ∈ Mm×n(D) we define d(A,B) = rank(A−B). We call
d the arithmetic distance. Matrices A,B ∈ Mm×n(D) are said to be adjacent if
d(A,B)=1. If A∈Mm×n(D), then tA denotes the transpose of A.
In the series of papers [4] - [11] Hua initiated the study of bijective maps on
variousspacesofmatricespreservingadjacencyinbothdirections. LetV beaspace
of matrices. Recall that a map φ : V → V preserves adjacency in both directions
if for every pair A,B ∈ V the matrices φ(A) and φ(B) are adjacent if and only if
A and B are adjacent. We say that a map φ : V → V preserves adjacency (in one
direction only) if φ(A) and φ(B) are adjacent whenever A,B ∈V are adjacent.
Hua’s fundamental theorem of the geometry of rectangular matrices (see [25])
statesthatforeverybijectivemapφ:Mm×n(D)→Mm×n(D),m,n≥2,preserving
adjacencyinbothdirectionsthereexistinvertiblematricesT ∈M (D),S ∈M (D),
m n
a matrix R∈Mm×n(D), and an automorphism τ of the division ring D such that
(1) φ(A)=TAτS+R, A∈Mm×n(D).
Here, Aτ = [a ]τ =[τ(a )] is a matrix obtained from A by applying τ entrywise.
ij ij
In the square case m=n we have the additional possibility
(2) φ(A)=T t(Aσ)S+R, A∈M (D),
n
where T,S,R are matrices in M (D) with T,S invertible, and σ : D → D is an
n
anti-automorphism. Clearly, the converse statement is true as well, that is, any
map of the form (1) or (2) is bijective and preserves adjacency in both directions.
Composing the map φ with a translation affects neither the assumptions, nor
the conclusion of Hua’s theorem. Thus, there is no loss of generality in assuming
that φ(0)=0. Then clearly, R=0. It is a remarkable fact that after this harmless
normalization the additive (semilinear in the case when D is a field) character of φ
is not an assumption but a conclusion.
This beautiful result has many applications different from the original Hua’s
motivation related to complex analysis and Siegel’s symplectic geometry. Let us
mention here two of them that are especially important to us. There is a vast
literature on linear preservers (see [16]) dating back to 1897 when Frobenious [3]
described the general form of linear maps on square matrices that preserve de-
terminant. As explained by Marcus [17], most of linear preserver problems can
be reduced to the problem of characterizing linear maps that preserve matrices of
rank one. Of course, linear preservers of rank one preserve adjacency, and there-
fore, most of linear preserver results can be deduced from Hua’s theorem. When
reducingalinearpreserverproblemtotheproblemofrankonepreserversandthen
1
2 PETERSˇEMRL
to Hua’s theorem, we end up with a result on maps on matrices with no linearity
assumption. Therefore it is not surprising that Hua’s theorem has been already
proved to be a useful tool in the new research area concerning general (non-linear)
preservers.
It turns out that the fundamental theorem of geometry of Grassmann spaces
[2] follows from Hua’s theorem as well (see [25]). Hence, improving Hua’s theorem
one may expect to be able to also improve Chow’s theorem [2] on the adjacency
preserving maps on Grassmann spaces.
Motivated by applications we will be interested in possible improvements of
Hua’stheorem. Thefirstnaturalquestioniswhethertheassumptionthatadjacency
is preserved in both directions can be replaced by the weaker assumption that it is
preservedinonedirectiononlyandstillgetthesameconclusion. Thisquestionhad
beenopenedforalongtimeandhasfinallybeenansweredintheaffirmativein[13].
Next,onecanaskifitispossibletorelaxthebijectivityassumption. Thefirstguess
might be that Hua’s theorem remains valid without bijectivity assumption with a
minor modification that τ appearing in (1) is a nonzero endomorphism of D (not
necessarily surjective), while σ appearing in (2) is a nonzero anti-endomorphism.
Quite surprisingly it turned out that the validity of this conjecture depends on
the underlying field. It was proved in [19] that it is true for real matrices and
wrong for complex matrices. And the last problem is whether we can describe
maps preserving adjacency (in both directions) acting between spaces of matrices
of different sizes.
Let us mention here Hua’s fundamental theorem for complex hermitian ma-
trices. Denote by H the space of all n × n complex hermitian matrices. The
n
fundamental theorem of geometry of hermitian matrices states that every bijective
map φ:H →H preserving adjacency in both directions and satisfying φ(0)=0
n n
isacongruencetransformationpossiblycomposedwiththetranspositionandpossi-
blymultipliedby−1. Here,againwecanaskforpossibleimprovementsinallthree
above mentioned directions. Huang and the author have answered all three ques-
tions simultaneously in the paper [12] by obtaining the following optimal result.
Let m,n be integers with m ≥ 2 and φ : H → H a map preserving adjacency
m n
(in one direction only; note that no surjectivity or injectivity is assumed and that
m may be different from n) and satisfying φ(0) = 0 (this is, of course, a harmless
normalization). TheneitherφisthestandardembeddingofH intoH composed
m n
with the congruence transformation on H possibly composed with the transpo-
n
sition and possibly multiplied by −1; or φ is of a very special degenerate form,
that is, its range is contained in a linear span of some rank one hermitian matrix.
This result has already been proved to be useful including some applications in
mathematical physics [23,24].
ItisclearthattheproblemoffindingtheoptimalversionofHua’sfundamental
theorem of geometry of rectangular matrices is much more complicated than the
corresponding problem for hermitian matrices. Classical Hua’s results characterize
bijective maps from a certain space of matrices onto itself preserving adjacency
in both directions. While in the hermitian case we were able to find the optimal
resultbyimprovingHua’stheoreminallthreedirectionssimultaneously (removing
the bijectivity assumption, assuming that adjacency is preserved in one direction
only, and considering maps between matrix spaces of different sizes), we have seen
abovethatwhenconsideringthecorrespondingproblemonthespaceofrectangular
1. INTRODUCTION 3
matrices we enter difficulties already when trying to improve it in only one of the
three possible directions. Namely, for some division rings it is possible to omit the
bijectivity assumption in Hua’s theorem and still get the same conclusion, but not
for all. In the third secion we will present several new examples showing that this
is not the only trouble we have when searching for the optimal version of Hua’s
theorem for rectangular matrices.
Let m,n,p,q be positive integers with p≥m and q ≥n, τ :D→D a nonzero
endomorphism, andT ∈M (D)andS ∈M (D)invertible matrices. Thenthe map
p q
φ:Mm×n(D)→Mp×q(D) defined by
(cid:2) (cid:3)
Aτ 0
(3) φ(A)=T S
0 0
preserves adjacency. Similarly, if m,n,p,q are positive integers with p ≥ n and
q ≥ m, σ : D → D a nonzero anti-endomorphism, and T ∈ M (D) and S ∈ M (D)
p q
invertible matrices, then φ:Mm×n(D)→Mp×q(D) defined by
(cid:2) (cid:3)
t(Aσ) 0
(4) φ(A)=T S
0 0
preserves adjacency as well. We will call any map that is of one of the above two
forms a standard adjacency preserving map.
Having in mind the optimal version of Hua’s theorem for hermitian matrices it
is natural to ask whether each adjacency preserving map between Mm×n(D) and
Mp×q(D) is either standard or of some rather simple degenerate form that can be
easily described. As we shall show in the third section, maps φ : Mm×n(D) →
Mp×q(D)whichpreserveadjacencyinonedirectiononlycanhave awildbehaviour
that cannot be easily described. Thus, an additional assumption is required if we
want to have a reasonable result. As we want to have an optimal result we do not
want to assume that matrices in the domain are of the same size as those in the
codomain, and moreover, we do not want to assume that adjacency is preserved in
both directions. Standard adjacency preserving maps are not surjective in general.
They are injective, but the counterexamples will show that the injectivity assump-
tionisnotstrongenoughtoexcludethepossibilityofawildbehaviourofadjacency
preserving maps. Hence, we are looking for a certain weak form of the surjectivity
assumption which is not artificial, is satisfied by standard maps, and guarantees
that the general form of adjacency preserving maps satisfying this assumption can
be easily described. Moreover, such an assumption must be as weak as possible so
that our theorem can be considered as the optimal one.
Inordertofindsuchanassumptionweobservethatadjacencypreservingmaps
are contractions with respect to the arithmetic distance d. More precisely, assume
that φ : Mm×n(D) → Mp×q(D) preserves adjacency, that is, for every pair A,B ∈
Mm×n(D) we have
d(A,B)=1⇒d(φ(A),φ(B))=1.
Usingthefacts(seethenextsection)thatdsatisfiesthetriangleinequalityandthat
for every positive integerr and every pair A,B ∈Mm×n(D)we have d(A,B)=r if
and only if there exists a chain of matrices A = A ,A ,...,A = B such that the
0 1 r
pairs A0,A1, and A1,A2, and ..., and Ar−1,Ar are all adjacent we easily see that
φ is a contraction, that is
d(φ(A),φ(B))≤d(A,B), A,B ∈Mm×n(D).