CombinatorialMatrixTheory(EncyclopediaofMathematicsanditsApplications)

Thisisthefirstbookdevotedtotheexpositionofcombinatorialmatrixtheory.Thissubjectconcernsitselfwiththeuseofmatrixtheoryandlinearalgebrainprovingresultsincombinatorics(andviceversa),andwiththeintrinsicpropertiesofmatricesviewedasarraysofnumbersratherthanalgebraicobjectsinthemselves.Therearechaptersdealingwiththemanyconnectionsbetweenmatrices,graphs,digraphsandbipartitegraphs.Thebasictheoryofnetworkflowsisdevelopedinordertoobtainexistencetheoremsformatriceswithprescribedcombinatorialpropertiesandtoobtainvariousmatrixdecompositiontheorems.Otherchapterscoverthepermanentofamatrix,andLatinsquares.Thefinalchapterdealswithalgebraiccharacterizationsofcombinatorialpropertiesandtheuseofcombinatorialargumentsinprovingclassicalalgebraictheorems,includingtheCayley-HamiltonTheoremandtheJordanCanonicalForm.Thebookissufficientlyself-containedforuseasagraduatecoursetext,butcompleteenoughforastandardreferenceworkonthebasictheory.Thusitwillbeanessentialpurchaseforcombinatorialists,matrixtheorists,andthosenumericalanalystsworkinginnumericallinearalgebra.