Matricial norms : some applications and open questions

Pub . Ma . UAB N° 22 No . 1980 Ac es VII JMHL MATRICIALNORMS : SOME APPLICATIONS AND OPEN QUESTIONS José Vi ó ia Dp o . de Ma ema£ica . Uni e sidade de Coimb a Abs ac - We men íon some heo e ical aspec s o ma icialno ms, including some ecen de elopmen s E obe , Deu sch, Baue , Ba ke , Meixne , Coimb a= . Then we e e se e alapplica ions o ma icial no ms : con e gence o i e a i e p ocedu es, weak con ac ion in ec o ial no m/ ixedpoin ques ions E obe l ; app oxima ion in -me ic spa ces ICoimb a : ; localiza ion o ze os o polynomials Eeu sch, Vi- ó ial ; bounds o la en oo s o lambda-ma ices E i ó ia : ; i- b a ing sys ems c limaco}Vi ó ia : ; lowe bounds o linea mappings E ode= . Finally, we include some open ques ions conce ning bo h heo e ical and p ac ical aspec s . 1 . Applica ions o ma icial nonns In his sec ion we e e some applica ions o p ope ies o ec o ial Ea icial : no ms . 1 .1 . Con e gence o linea i e a ions One u ilizes p ope ies o ma icial no ms o con ol he con e gence o i e a i e p ocedu es-speed and e o . . Block-i e a ions : Gauss-Ssidel, Ja- cobi, o e - elaxa ion . * The wo k o his pape was suppo ed, a di e en s ages, by Fundagáo Calous e Gulbenkian (Lisboa, Po ugal), Uni e sidade de Mapu o (Mozambique) and Ins i u o Nacional de In es iga gáo Cien í ica (Lisboa, Po ugal) . 1 .2 . "Sé ie-pa allee" i e a i e me hods . One cons uc s algo i hms o linea i e a ions i ed o au oma ic calcula ions "in pa allel" - se e al a i hme ic uni s, wo king simul aneously, b anched o a common memmo y . These algo i hms can be placed a an in e me dia y le el be ween he Gauss-Seidelme hod - sequen ial - and he Jacobime- hod - pa allel . 1 .3 . Weakcon ac ion in ec o ial noim/ ixed poin p oblems Pe on-F obenius heo y o ma ices pa i ioned in o blocks . 1 .4 . Con e gence o ma ix sequences The e a e esul s, using (scala ) no ms, on con e gence o ma ix se quences . Such esul s can be gene alized by using ec o ial no ms . 1 .5 . Bounds o he ze os o polyncmials One akes he companionma ix  C  o a polynomial  p(z) =a o zn + a1zn-1 + + . . .+a n-l z+ an ,a iE C, (i=0,1, . . ., n)  and cons uc s se e al adequa ed ma i- cial no ms ela ed o ma ix C . So we ge uppe bounds o he spec al a- dius o ma ix C, i .e . o he absolu e alues o he oo s z o p(z) . 1 .6 . Bounds o he la en oo s o lambda-ma ices -  -  We conside he lambda-ma ix M(X) =Ion+A1 an-1+ .. .+An-l a+An, whe e Ai , I EM(p C) p , (i=l, . . . . n) .  We  need o  know he  ze os o de M (a) = I IX n+Al an-!+ . . . . .. +An-l a+A n 1 =0,  ha is o soy, we need o know he la en oo s  a  o he lambda-ma ix . M(a) . The calcula ion o he la en oo s may become a o mi- dable ask, his depending on n and p . Ins ead o calcula ing hem, we look - Po localiza ion domains o he la en oo s . We associa e o he lambda-ma ix i s block-companion ma ix M . The eigen alues o block-companion ma ix M a e he la en oo s o he lambda- -ma ix . Then we cons uc se e al ma icial no ms ela ed o he ma ix M, and ge uppe bounds o IXI . Inciden ally, we ob ain known (and new) esul s o ze os o (scala ) polynomials . 1 .7 . App oxima ion in -me ic spaces Gene aliza ions o -me ic spaces i seudo-me ic spaces : o esul s in me ic spaces . Gene aliza ion o New on's me hod o supe - - me ic spaces, he ec o ialme ic being induced by a ec o ial no m . 1 .8 . Lowe bounds o linea mappings Gi en a ( egula ) ec o ial no m T o o de k un k n . Lowe bound o a ma ix Co :pe a o ] A eM n ~ n (K)  wi h espec o he ec o ial no m T is n a ma ix NEMk,k(R)  such ha  VxeK , N 4' (x) = (Ax) . We can apply hese lowe bounds o he es ima ion (in ec o ial no m) o , he ob ained app oxima ion o he solu ion o a linea sys em . blocks . 1 .9 . Eigen alues in block-pa i ioned ma ices Gene aliza ion o Ge schgo in esul s o ma ices pa ioned in o 1 .10 . Vib a ing sys ems Fo s udying he mo ion equa ions, one can use (classical) Lag ange me hod . The solu ion o such equa ions will ep esen a sinusoidal mo ion, so we lock o solu ions o an equa ion o ype A~+Bp+Cp = P . Fo sol ing hese equa ions we need o know he la en oo s o he lambda-ma ix  M()=AX 2 + +Ba+C . Bu in ce ain si ua ions we do no need o calcula e he la en oo s o lambda-ma ix M(X) . I su ices o know a localiza ion domain o he la- en oo s . In his case a e well sui ed he esul s e e ed o in 1 .6 . 2 . Open ques ions In his sec ion we men ion some seemingly open ques ions, in ol ing ma icial no ms : 2 .1 . De ia ion o a ma ix omsingula i y 2 .2 . De ia ion o a ma ix om noimali y 2 .3 . Spec al a ia ion o wo ma ices 2 .4 . Gene aliza ions o a undamen al inequali y un ma icial no ms (*)  p (A) <p [,D(A)7  ,  p (M)  :  Spec al adius o a ma ix  Mi E M)  : ma icial no m o a ma ix  M . We aise he ollowing ques ions : a) I is possible o conside a ela ion "simila " o (*) in he con ex o ma ices whose elemen s a e in e als? b) How "seems" ela ion (*) i he elemen s o ma ix A be- long o a p .i . . (p incipalideal aing)? u1 ."2 . . . . ."k ha c) Le  X lI x2  .. .x n  be he eigen alues o  A EM n .n (K)  and be he eigen alues o i s ma icial no m (D(A) CMk k (R) . Supp9se 1 x 111 1X21 > . -_ I X, I  and  l"11 1I"21 1 . . . 2 !" k l F om (*) we ha e Ixll = P(A) <P (D(A)) = lull Wha can we say abou ma ices such ha I x2 1_ I" 2 1  1x 3 1=1" 3 1  . Rema k A comple e sion o his pape will appea elsewhe e, PRINCIPAL REFERENCES e c . ? A . BODE  - Symme ien und An isymme ien de Robe 'schenQuasio dnung in Ma izen aumen, Lin . Alg . and Appl . 23 :87-99 (1979) . J . CLIMACOj J . VITóRIA - Sis emas Vib a ó ios, ma izes-lambda e no mas ec- o iais . I Encon o de Físicos e Ma emá icos, Coimb a, 11-12 Ou ub o, 1979 . E . COIMBRA - Ap oximagóes em espagos -mé icos . Tese . 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