矩阵空间的保持问题

Preface

Notation

Chapter1Introductiontopreserverproblems

1.1Definitions

1.2Typicalcategoriesofpreserverproblems

1.3TechniquesofPreserverProblems

1.4Organization

Chapter2Preserversofthesmallestnonzerorank

2.1Additiverank-1preserversbetweenspacesofrectangularmatrices

2.2Additiverank-1preserversbetweenspacesofsymmetricmatrices

2.3Additiverank-1preserversbetweenspacesofHermitianmatrices

2.4Additiverank-1preserversfromaspaceoftriangularmatricestoaspaceofsquarematrices

2.5Additiverank-2preserversbetweenspacesofalternatematrices

2.6Invertiblelinearrank-1preserversonspacesoftracezeromatrices

2.7Invertibility(characteristicpolynomial,determinant)preservers

2.8RemarksonChapter2

Chapter3Additiverank-additivitypreserversandapplications

3.1Additiverank-additivitypreserversonspacesofrectangularmatrices

3.2Additiverank-additivitypreserversonspacesofsymmetricmatrices

3.3Additiverank-additivitypreserversonspacesofHermitianmatrices

3.4Additiverank-additivitypreserversonspacesofalternatematrices

3.5Applicationsofadditiverank-additivitypreservers

3.6RemarksonChapter3

Chapter4Additivepreserversofrankequivalenceandapplications

4.1Mainresults

4.2Proofsofmainresults

4.3RemarksonChapter4

Chapter5Linearsquare-zeropreserversonspacesofsquarematrices

5.1Mainresults

5.2Proofsofmainresults

5.3RemarksonChapter5

Chapter6Preserversofk-power/k-potent

6.1Additivepreserversofidempotencebetweenspacesofsquarematrices

6.2Injectiveadditivepreserversofidempotenceonspacesoftriangularmatrices

6.3Linearpreserversofidempotencebetweenspacesofsymmetricmatrices

6.4Severalproblemsrelatedtok-powerandk-potentpreservers

6.5Preserversofk-power

6.6Preserversofk-potentmatrices

6.7Jordanhomomorphismsbetweenofringsofsquarematrices

6.8Preserversofidempotentrelation

6.9RemarksonChapter6

Chapter7Preserversofinversesofmatrices

7.1Linearinversepreserversbetweenspacesofmatrices

7.2Additiveinversepreserversbetweenspacesofmatrices

7.3RemarksonChapter7

Chapter8Preserversofgeneralizedinversesofmatrices

8.1AdditivepreserversofM-Pinversesofmatricesoverfields

8.2Additivepreserversof{1,2}-inversesofmatricesoverfields

8.3Additivepreserversofgroupinversesofmatricesoverfields

8.4StronglinearpreserversofMoore-Penroseinversesofmatricesoverthetwo-elementBooleanalgebra

8.5CovariantoperatorpairsonM-PInversesofmatricesbetweenrealfinitedimensionaldivisionalgebras

8.6RemarksonChapter8

Chapter9Linearpreserversofinvolutorymatrices

9.1Mainresults

9.2Proofsofmainresults

9.3RemarksonChapter9

Chapter10Additiveadjointpreservers

10.1Mainresults

10.2Proofsofmainresults

10.3RemarksonChapter10

Chapter11Preserversofmultiplication

11.1Homomorphismsfromonegenerallineargrouptoanother

11.2Homomorphismsfromamultiplicativematrixsemigrouptoanother

11.3Automorphismsofmultiplicativesemigroupsofuppertriangularmatrices

11.4RemarksonChapter11

AppendixAFittingdecompositionofamatrix

AppendixBCanonicalformofarank-additivematrixpair

AppendixCCanonicalformofamatrixtriple

AppendixDAdjointmatrices

Bibliography

Index

　　Oneoftheveryactiveresearchareasinmatrixtheoryisthestudyofpreserverproblems,includinglinearpreserverproblems,additivepreserverproblemsandmultiplicativepreserverproblems,whichconcerntheclassificationofmapsonmatricesoroperatorsthatpreservecertainspecialproperties.　　Theaimofthisbookistoprovideanintroductiontocurrentadvancesofpreserverproblemsonmatricesandpresentthebasicmethodsofstudyingpreserverproblems.Inordertoensurethatthecontentisself-contained,wetooksomeconclusionsandtheirproofsfromotherbooksorpapers,whichwillbeconvenienttothereaders.Itisalsointendedtoofferacollectioaofreferencesonpreserverproblemsonmatrices.Thisbookisespeciallyappropriateforadvancedundergraduatesmajoredinmathematics,andalsointendedforgraduatestudentsandresearchlevelmathematicians.Itishopedthatthebookwillbesuitabeforpostgraduatetouseorasareference.Thereadersarerequiredtofamiliarizewithmatrixmethodsandtocomprehendsomebasicconceptsinalgebra,suchaslinearspace,(semi-)group,(semi-)ring,homomorphism,isomorphism,etc.