Normal matrices with entries from an arbitrary field of characteristic not-equal 2

Abstract

Le t F be a n a rbitrary fi e ld of c ha racte ri sti c 7" 2 a nd le t co nj ugati on in F be de fin e d b y a n a rbitra ry involutory a uto morphi s m (poss ibl y th e ide ntit y ma ppin g). A matrix with e ntri es fro m F is norm al if it co mmut es with its co nju gate tra ns pose. Seve ra l co nditi o n whi c h a re equi va le nt to norm alit y wh e n F is th e co mpl ex fi e ld are pro pe rl y nes te d whe n F is a n a "b itrary fi e ld. Of th ese con diti ons, no rm a lit y is th e wea kes t a nd unit a ry di ago nali zabi lit y is th e s tronges t. Wh e n th e unde rl yin g fi e ld is c losed th e unit ari ly dia go na lizabl e matri ces a re th ose whi c h possess a s pectra l re prese nt a ti o n with He rmiti a n ide mpote nt s. These matri ces may be ch arac teri ze d in ter ms of th eir EP pro pe rti es. A m atri x is ind ecomposa bl e if it has a s in gle e le mc nt a ry di viso r of the form (X-S)i. For i > 1, normal indecomposable matrices do not exist whe n F is the c ompl ex fi eld. H o we ve r th ere exi st fi elds for wh ic h normal indecomposa bl e ma tri ces of all finit e orde rs ex is t. A matri x is a no rm al indeco mposa ble matri x if a nd onl y if it ca n be ex pressed as r(B) wh e re r(x) is a polyno mi a l s uc h th at r'(O) 7" 0 a nd B is a ma trix ha vin g th e sin gle e le me nt a ry divi~o r Xi and sati sfyin g th e e qu ati o n B*= sB [o r so m e sca la r s. Wh e n th e in vo lut ory a utomorphi s m o[ F is th e id e ntit y m a p pin g in deco mposable nor ma l matri ces of e ve n o rder ex ist if a nd o nl y if th e vec to r s pace F" is hype rb oli c , and in thi s case th e ma tri ces a re sy mm e tri c. Moreov e r , ind eco mposabl e no rm al matri ces of odd o rd e r ex is t if an d "nl y if F " is th e o rth ogo na l SUm o[ a hype rb oli c s pace a nd a o ne dim e ns iona l s pace. a nd in thi s case both sy mm e tric an d no nsymme tri c ind eco mposabl e norm al ma t ri ces exis t.