Chapter 9 Jordan Block Matrices
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1 Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste. Here we cosder a matrx smple f t s as close as possble to dagoal. Ufortuately, ot every matrx s smlar to a dagoal oe. We wll troduce ad study the ext best thg, the Jorda block matrx. But frst we vestgate the optmstc case: there exsts a bass R such that A R (T ) s a dagoal matrx,.e. a f. What s so specal about ths bass? Let us deote vectors of R by v,v,... v. The, for each we have T(v ) a v. Defto 9.. A scalar s called a egevalue of T ff there exsts a ozero vector v such that T(v) v. Every such vector s called a egevector of T belogg to. Theorem.. The matrx R (T ) s dagoal ff R s a bass cosstg etrely of egevectors of T. It seems that egevalues ad egevectors have a mportat part to play ths theory. How do we fd egevalues ad egevectors for a operator? We smply follow the defto, except that we use the matrx of T a bass R of our choce, place of T tself. For example f R s the stadard bass for F ad A s the matrx of T R the T(x)Ax. The equato T(x) x s the equvalet to AxxIx. Ths yelds (A-I)xΘ. Thus s a egevalue for T (or for A) ff the homogeeous system of equatos (A-I)xΘ has a ozero soluto (besde the zero soluto, whch s always there). Every such soluto s the a egevector of T belogg to the egevalue. Sce solutos of a homogeeous system of lear equatos form a vector space we get that all egevectors belogg to a partcular egevalue (together wth the zero vector, whch s ot a egevector tself) form a subspace of F, called a egespace. Nozero solutos exst f ad oly f the rak of A s less tha - the umber of colums of A. Ths happes f ad oly f det(a-i). It s easy to see that the fucto det(a-i) s a polyomal of degree varable. It s called the characterstc polyomal of the operator T (ad of the matrx A). Hece we have proved
2 Theorem 9.. A scalar t s a egevalue for A ff t s a root of the characterstc polyomal of A. Theorem 9.. If,,..., are parwse dfferet egevalues for T ad for each, L {v,,v,,...,v,k() } s a learly depedet set of egevectors belogg to the L L... L s learly depedet. Roughly speakg, the theorem states that egevectors belogg to dfferet egevalues are learly depedet. Proof. Frst, suppose that every L s a oe-elemet set. For smplcty we wll wrte L {v }. We prove our case by ducto. For there s othg to prove. Suppose the theorem holds for some ad cosder the codto + a v Θ. Applyg T to both sdes we get + + at ( v ) a v Θ, whle scalg both sdes by + yelds + + a + v Θ oe from the other we get ( + ) av ( + ) av Θ because for +,. Subtractg - +. By the ducto hypothess, ths mples ( - + )a for,,...,. Sce the egevalues are assumed to be parwse dfferet, we get a a... a. Ths ad + a v Θ mply that a + v + Θ, hece a +. To coclude the proof suppose that L k() ad a, v, Θ. For each deote k ( ) w k ( ) a, v,. Each vector w, beg a lear combato of egevectors belogg to the same egevalue s ether equal to Θ or s tself a egevector belogg to If some of them, say w (),w (),... w (p) are fact egevectors the k ( ) a, v w w ( t) Θ s a, t lear combato of egevectors belogg to dfferet egevalues. Hece, by the frst part p of the proof, all coeffcets of the lear combato p w t ( t) Θ are zeroes, whle everybody ca clearly see that they are oes. Ths s mpossble sce every feld. Hece all w Θ. Sce each set L s learly depedet, all a, are equal to.
3 Example 9.. Fd a dagoal matrx for T(x,y,z) (x+y-z,x+y-z,x+y+z). The matrx A of T wth respect to the stadard bass of R s. The characterstc polyomal of A s A () (-) (-). For we get A-I A-I. The rak of A-I s obvously ad we ca easly fd a bass for the soluto space, amely {(,-,), (,,)}. For we get A-I A-I. The rak of A-I s ad we ca chose (,,) as a egevector belogg to. The set R {(,-,), (,,),(,,)} s a bass for R ad R (T). Ufortuately, ot every matrx ca be dagoalzed. For some matrces (operators) there s o bass cosstg of egevectors. I those cases the ext best thg s the Jorda block matrx. Defto 9.. A block matrx s a matrx of the form B s B B A where each B s a square matrx, the dagoal of B s a part of the dagoal of A ad all etres outsde blocks B are zeroes. Defto 9.. A Jorda block of sze k ad wth dagoal etry s the k k matrx B Defto 9.. A Jorda block matrx s a matrx of the form
4 J B B where each B s a Jorda block. B s Example 9.. J s a Jorda block matrx wth three Jorda blocks. e of sze wth dagoal etry, oe of sze wth dagoal etry, ad oe of sze wth dagoal etry. Theorem 9.. (Jorda) Suppose T:F F ad T has egevalues (coutg multplctes). The there exsts a bass R for F such that J R (T) s a Jorda block matrx. Proof substtute. We wll descrbe (workg backwards) how to costruct such a bass R {v,...,v } studyg propertes of ts (hypothetcal) vectors. Suppose that the sze of B, the frst Jorda block, s k ad the dagoal etry s. The, by the defto of the matrx of a lear operator wth respect to R ad by the defto of a Jorda block matrx, we get T(v ) v + v v v,.e. v s a egevector for. Ths ca be expressed dfferetly as (T-d)v Θ. Sce the secod colum of J cossts of, ad all zeroes, we get that T(v )v +v. I other words, (T-d)v v. Extedg ths argumet we get that for each v from the frst k vectors from R we have (T-d)v v -, except that for v w have (T-d)v Θ. Vectors v,..., v k are called vectors attached to the egevector v, of the order,,...,k-, respectvely. Now the structure of R becomes clear. It cossts of several buches of vectors. The umber of the buches s the umber of Jorda blocks J, each buch s lead by a egevector ad followed by ts attached vectors of orders, ad so o. It s all very ce the hdsght, kowg what the matrx J looks lke, but how do we actually fd those vectors? r at least how do we fd the matrx J? Well, kds, here comes the story. () Gve a lear operator, you should fd ts matrx A a bass of your choce, most lkely your choce wll be the stadard bass S. () Havg foud the matrx A, fd all egevalues solvg the characterstc equato of T,.e. det(a-i).
5 () For each egevalue fd the umber of Jorda blocks wth ths partcular egevalue as the dagoal etry. Ths s, of course, equal to the maxmum umber of learly depedet egevectors belogg to, other words the dmeso of the soluto space of the system of equatos (A- I)XΘ, other words -rak(a- I). () Calculate szes of Jorda blocks wth as the dagoal etres usg the formula: rak(a- I) k -rak(a- I) k+ s the umber of -Jorda blocks of the sze at least k+. It follows from the fact that raks of matrces (A- I) k ad (J- I) k are the same (because the matrces are smlar) ad that L L L, ad wth each ext power of the matrx, the le of oes goes oe posto to the rght losg oe, utl there s othg left to loose. Hece raks of all used-to-be- blocks of the matrx J- I go dow by oe wth each cosecutve multplcato by J- I, except for those, whose raks have already reached. The blocks correspodg to other egevalues reta ther raks. Hece we have: -rak(a- I) Jorda blocks of all szes, rak(a- I)- rak(a- I) Jorda blocks of sze at least, rak(a- I) - rak(a- I) Jorda blocks of sze at least, ad so o. () Repeat steps ()-() for each egevalue,,...,. To fd the bass R, we fd frst attached vectors of the hghest order, ad the trasform them by A- I tll we get egevectors. Assumg that the largest order of a attached vector s k-, we fd the vector v k choosg a soluto to (A- I) k XΘ that satsfes (A- I) k- X Θ. The we put v k- (A- I)v k, v k- (A- I)v k-,..., v (A- I)v. If we have more tha oe Jorda block the we have to take care whe we choose our attached vectors so that they ad ther egevectors are learly depedet. Ths may be trcky ad you must be extra careful here.
6 Problem. Fd a bass R for R such R (T) s a Jorda block matrx J, where T(x,y,z,t)(-x-y-z+t,x+y+z-t,x+z-t, x-y+z) Soluto.Frst we form A - the matrx for T the stadard bass of R. bvously A. Next we calculate det det( I) A (r -r ) ( (c +c ) ( ( (r +r,r - r ) ( det ) ( (-). Now we must calculate rak(a-i) rak (r +r,r +r,r +r ) rak (r -r,r -r ) rak. Ths meas that J has two blocks wth dagoal etres, but szes of the blocks may be ad or ad. Now we must calculate raks of matrces A-I, (A-I) ad so o. It turs out that (A-I) s the zero matrx, so ts rak s. By part () of our algorthm we get that J has blocks of sze at least each, that s J has blocks of sze by. Hece J. Thus our bass R cossts of two egevectors v ad v ad ther attached vectors v ad v. The attached
7 vectors are solutos of (A-I) XΘ, that do ot satsfy (A-I)XΘ. Sce (A-I) s the zero matrx, the frst system of equatos s trval (ΘΘ), so the oly codto v ad v must satsfy s (A-I)X Θ. We ca choose v (,,,) ad v (,,,) gettg v (-,,,) ad v (-,,,-). These vectors form the colums of the chage-of-bass matrx P such that J P - AP,.e. P.
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