On block-quasi-tridiagonal matrices

Pub . Ma . UAB N° 20 Se . 1980 Ac es . VII JMHL ON BLOCK-QUASI-TRIDIAGONAL MATRICES José Vi ó ia Dp o . de Ma emá ica Uni e sidade de Coimb a Abs ac - We in end o in e (and o s udy he eigen alues o ) block-quasi- - idíagonal ma ices, We also men ion ways o handling he p o- blem o eigen alues o a block-quasi- idiagonal ma ixand we ob ain uppe bounds o he spec al adius o á (ce ain) block- -quasi- idiagonal ma ixwhich a ises in he disc e iza ion o pa ial di e en ial equa ions o ellip ic ype, sel -adjoin case . 1 . Abou he in e inn o block-quasi- idiagonalma ices 1 .1 - In his sec ion ws in e es us o in e ing block-quasi- i- diagonal ma ices, ha is o say, ma ices o he o m A n-l,n A nn * 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 igagáo Cien í ica (Lisboa, Po ugal) . A 11 A12 00 . . . 0 0 0A 1 n T : _ / A 21 A 22 A 23 0 .. . 0 0 0 0 00 0 0 . . . 0 A n-l,n-2 A n-l,n - 1 Anl 0 0 0 0A n,n-1 whe e ell blocks a e o he-same o de and commu e in pai s . Such ma ices a ise in he disc e iza ion o ellip ic pa ial di e en ial equa ions . We need a esul in ol ing a ma ix ob ained om T by aking i s de e minan conside ing he (commu ing) blocks as elemen s . Le AT : = de T be such a ma ixi AT is he o mal de e minan o T . I is known ha de A T =de T . This esul allows us o manipula e only ma ices o low o - de , when sol ing la ga sys ems o linea equa ions and, like in ou case, in e ing la ge block-ma ices . 1 .2 - To in e he ma iz T we shall use an hyb id me hod : classi= cal pa i ioning in ou blocks plus a ecu ing p ocedu e . Le us pa i ion and no e as ollows is 228 T= I A ln 10 '  i A n1 0 0 . . .0 An .n-1(Ann whe e P ie a block- idiagonal ma ix . I is known ha he in e sa o a ma iz pa i ioned in ha manne wi h he same pa i ioning and whe e K : =P -1 - P -1 QM . M : _ - NRP -1 , L : _ - P -1 QN  and  N : = (S-RP-1Q)-l . he j h block-column wi h need only o in e wo ma ices : P  and  (S-RP -1 Q) . So we a la ga ma iz P , wi h commu ing blocks . Fo achie ing his, P -1 =(X ij ) pa i ioned commu e in pai s, om P by eplacing Then, i we pu A : =de P and T-1 : _  - K Í -N- LM  ~ In his way we ha e o in e one can use a ecu ing p ocedu e . We ob ain a ma ix in he sama way as P and whe e he blocks X ij also Le us deno e by P I(i, ) he ma iz ob ained e hé ma iz i ) . = P-Í-Q- R I S .  1 i h block-line . --------------------1A4_1_-- 1 ij : =de PI(i,j)  -  (i,j=1,2 . . . . . n-1), we ob ain Xi .)  = A-1  nij  ,  (i,j=1 .2-  n-1) 1,3 - As we ha e sean, o in e ing a block-quasi- idiagonal ma ix we need only o in e wo ma ices : S-RP -1 Q and A : =de P . Bu , in p a i- ce, we in e wo ma ices o he same o de , ins ead o in e ing a low o de ma ix S-RP -1 Q and a high o de ma ix P . We ema k ha he ma ices S-RP -1 Q and A ha e he o de o he o iginal blocks A¡J, 2 . Abdu he eigen alues o a (ce ain) block-quasi- idiagonal ma ix In his sec ion we in e es us o he eigen alue p oblem in block- -quasi- idiagonal ma ices . He e we ge uppe baunds o he absolu a alue o he eigen alues by using a ma icial no m : Uppe bounds o he spec al adius o a block-quasi- idiagonal má ix, which a ise in he disc e iza ion o pa ial di e en ial equa ions o ellip ic ype, sel -adjoín case . Gi en he ma ix A B 0 0 .  0 0 B BAB0 ... 000 08AB, . . 000 . ... ... . . . . . . . . 0 000 . .., BAB B000 . . .0BA (1) Fo  B = (s i .) e M s (K), K=R (o  C), we le J we look o uppe bounds o p(T), whe e p(T) is he spec al adius o he ma ix T, ha is o say, p(T) : =Max IX (T) I, whe e a (T) is any eigen a- lue o T . We ake he (scala ) no m  II " i11 1) , (1=1,°°), o each block, so ob ai ning a ma icial no m  M i (T) .  I is known, ha  p(T) _<p(M i (TM , (i=1,°°) . And i is also known ha  p (A) < II A II J ,  (j=l . .°°), o any ma ix  A, and any (subo dina e) ma ix no m 11 -Il i  s J1$111 : = Max  { E Isij1 , IIBII .== Max  ( E 1Bij l} . j=1,2, . . .,s i=1  i=1,2, . . ., j=1 Hence we ha e Rema k p(T)<¡¡Alli* 2 11Bili , (i=1,°°) a e y simple uppe bound o he eigen alues o T . The comple e e sion o his pape is o appea in "Re is ada Uni- e sidade de Coimb a" . PRINCIPAL REFERENCES TENEUR, J . - É ude c i ique de mé hodes apides pou la solu ion de Z'équa- ionde Poisson dansun ec angle . (P ojec de in d'é udes) . Ins i u Poly hécnique de G enoble - Ma h . Appliquées . G enoble . 1970 . VITóRIA, J . - Ma ices pa i ionnées en blocscommu a i s . Gaz . Ma . (Lisboa), 117-120 : 49-57 (1970), MR 49 # 3281 Zb 278, 15008 . In e sion o ma ices pa i ioned in commu ingblocks . Re .Cienc .Ma . 4 : 29-32 (1973) . MR 52 # 31871 Zb 322, 15009 . Ma izes de blocos comu a i os . Tesa . Uni . Lou engo-Ma ques C now Mapu o : , Mozambique . 1974 . No mas ec o iais de ec o es e de ma ices . (Repo ) . Uni e sidade de Lou enQo-Ma ques í- now Mapu o 7, Mozambique . 1974 . G E O M ETRIA  A L GEBR A I C A Clas . A .M.S .  13,14 Mesap esidencial : Ped o Abellanas, M . Luisa No onha, F ancisco Pé ez Mo- naso , Edua doCasas