JORDAN CANONICAL FORM AND ITS APPLICATIONS

Transcription

1 JORDAN CANONICAL FORM AND ITS APPLICATIONS Shivani Gupta 1, Kaajot Kau 2 1,2 Matheatics Depatent, Khalsa College Fo Woen, Ludhiana (India) ABSTRACT This pape gives a basic notion to the Jodan canonical fo and its applications. It pesents Jodan canonical fo fo linea algeba, as an alost diagonal atix, and copaes it to othe atix factoizations. It also shows that any squae atix is siila to a atix in Jodan canonical fo. Soe backgound on the Jodan canonical fo is given. Keywods : basis,eigen Vectos, eigen values, Jodan chains,nilpotent. I. INTRODUCTION The Jodan Canonical Fo (JCF) is undoubtably the ost useful epesentation fo illuinating the stuctue of a single linea tansfoation acting on a finite-diensional vecto space ove C (o a geneal algebaically closed field. The Jodan Fo Let T L(V) and let λ be an eigenvalue of T. Fo a positive intege, the subspace E (λ) = ke (T-λI) is called the genealized eigen space of ode associated with λ. E 1 (λ) is an eigenspace associated with λ, since V is finite diensional, thee is a positive intege p such that {0} = E 0 (λ) E 1 (λ). E p (λ) = E p+1 (λ) =.. An eleent x E (λ) \ E -1 (λ) is called a genealized eigenvecto of T of ode coesponding to λ. Clealy if x is a genealized eigenvecto of ode then (T-λI) x is a genealized eigenvecto of ode -1. A sequence of non zeo vectos x 1,., x k is called a Jodan chain of length k associated with eigenvalue λ if T x 1 = λ x 1, Tx 2 = λ x 2 + x 1.. T x k = λ x k + x k-1. A Jodan chain consists of linealy independent vectos. Poof: Let x 1,.,x k be a Jodan chain fo T associated with eigenvalue λ. Assue that α 1 x α k x k = 0 and that is the lagest index such that α 0. Clealy 1. Wite then x = x i and opeate (TλI) -1 on both sides to get x 1 =0, a contadiction. The length of a Jodan chain cannot exceed the diension of the space and the subspace geneated by a Jodan chain is T- invaiant. If B={x 1,., x k } consists of a Jodan chain, W= B and T ' is the Linea opeato on W induced by T then 1583 P a g e

2 λ 1.. [T '] B = λ This atix is called the Jodan block of size k associated with eigenvalue λ and we denote it by J k (λ). Note that J k (λ) λi k is a nilpotent atix of ode k. If V has a basis which is disjoint union of Jodan chains fo T, then the atix epesentation of T with espect to this basis is a block diagonal atix with Jodan blocks on the diagonal. This basis is called a Jodan basis of V fo T, and the coesponding atix epesentation a Jodan canonical fo fo T. Existence of Jodan canonical fo: If the chaacteistic polynoial of T splits ove K, then V has a Jodan basis fo T. Poof: Fist assue that T is nilpotent. We pove by induction on n. If T = 0 o in paticula n=1, then any basis of V is a Jodan basis. Suppose that T 0 and the stateent holds fo all vecto spaces ove K of diension less than n. Since T is a nilpotent, W = i T is a pope T-invaiant subspace of V. Let T ' be the estiction of T on W. Then T ' L (W) and di W n, by induction hypothesis W has a Jodan basis B' fo T '. Let B' =, a disjoint union of Jodan chains, that is B i = {x i1,.,x ini } and T ' x i1 = T x i1= 0 and T ' x ij = T x ij = x i j-1 fo j 2(1)n i, (1)k. Now x 11,, x k1 a e linealy independent vectos of ke T. Extend it to fo a basis of ke T:{x 11,.., x k1, y 1,, y q }, q 0. Next each x ink W, choose x ink+1 such that T x ink+1 = x ink. Now wite B=, whee B i = B' i {x i n i+1 } fo (1)k, and B k+i = {y i }, fo (1)q. We now show that B is a basis of V. Clealy = B ' B +k +q = di ket+ di it = div = n. Next if = 0 Whee α ij K and β K. Then opeating T on both sides we have = 0, and so =0 fo j 2(1) n i +1, (1)k. Thus = 0 andv which iplies that fo (1)k, and fo 1(1)q as {x 11,.., x k1, y 1,, y q} is a basis fo ke T. Finally if T is an abitay then it follows that the inial polynoial of T is of the fo (x- λ 1 ) 1. (x- λ k ) k, whee λ 1,., λ k ae distinct eigen values of T. By piay decoposition theoe V= V 1 V k whee V i =ke (T- λ i ) i a T- invaiant subspace. Let T i be the estiction on V i. Then T i L (V i. ) and S i = T i - λ i I 1584 P a g e

3 is a nilpotent opeato on V i. Theefoe V i has a Jodan basis B i fo S i and hence fo T i associated with eigenvalue λ i. Hence B is a Jodan basis fo T. Let B be a Jodan basis of V fo T. Then the nube of genealized eigen vectos of T coesponding to eigenvalue λ and of ode up to s is di ke (T-λI) s Poof. Let B = be a Jodan basis and is union of disjoint Jodan chains: B i : {x i1,x i2.,x ini }, (1). Let B 1 B 2,...B d be all the Jodan chains coesponding to λ. Fo a positive intege p define We pove by induction on s that ke(t-λi) s has a basis consisting of nonzeo eleents of the set { : (1) l, j 1(1)s }.This will clealy pove the stateent. Since{x i1,x i2.,x li } is a linealy independent subset of ke(tλi). Thus to pove the induction hypothesis fo s = 1 we need to check that this set is actually a basis of ke(t - λi). lf v= ke(t - λi),then 0= (T - λi)v l = il1 il1 = l ( Theefoe =0 fo j 2(1) i, (1)l, and fo j 1(1) i,i l+1(1). Hence v= and {x 11,x 21.,x l1 } is a basis of ke(t-λi). Now assue fo s. We now show that ke(t-λi) s+1 has a basis consisting of nonzeo eleents of { : (1)l,j 1(1)s+1}. Since the nonzeo eleents of this set ae aleady linealy independent, we only need to veify that this set spans ke(t-λi) s+1. Let v= ke(t - λi) s+1.then (T-λI)v ke(t-λi) s = : (1)l, j 1(1)s Since (T-λI)v= l l (, We have =0 fo j 1(1) i,i l+1(1) and also that =0 fo j s+2(1) i, (1)l,wheneve i > s+1. Hence l v= value λ is. The nube of Jodan chains fo T of length associated with eigen 1585 P a g e

4 2 di ke(t-λi) - di ke(t-λi) +1 - di ke(t-λi) -1, ank ke(t-λi) +1 + ank ke(t-λi) -1-2ank ke(t-λi) o Poof. The nube of Jodan chains fo T of length at least M associated with λ is exactly the nube of genealized eigenvectos of T of ode coesponding to λ which appea in a Jodan basis. By above esult this is equal to l = di ke(t-λi) - di ke(t-λi) -1.Theefoe the nube of Jodan chains fo T of length exactly equal to associated with λ is l - l +1.let T be a linea opeato on V and let c T (x): (x-λ 1 ) n1 (xλ k ) n k whee λ 1,.. λ k ae distinct eigenvalues of T. Then the atix epesentation of T with espect to jodan basis is : J= diag(j 1,...,J k ), whee fo each (1)k,J i =diag (J (i,1) ( λ i ),. J (i, i) ( λ i ) ).We ode the size of these Jodan blocks such that (i,1).. (i, i ).Such a atix J is the Jodan canonical fo o siply the Jodan fo of T. Note that fo each eigen value λ i the nube i and (i,1),..,(i, i ) ae uniquely deteined by T. Fo each eigenvalue λ i the nubes i is the geoetic ultiplicity of λ i and (i,1)+..+(i, i )= n the algebaic ultiplicity of λ i. Also it is easy to veify that each J i is such that J i - λ i I ni, is a nilpotent of ode (i,1) Hence the inial polynoial of T is (x-λ 1 ) (11) (x-λ k ) (k1). Now we show by diect atix ultiplications that how an nxn atix can be tansfoed to its Jodan fo'. Let J = J n (0). Then J t J = I - E 11. Poof: J = E E n-1n. J t = E E n n-1. Thus J t J = (E E n n-1 )( E E n-1n ) = E E n n = I - E 11. Let J (0) = J, α C n and B C n x.n. Then Poof: Diect ultiplication of the atices on the left side and using that Je i+1 = e i, gives the ight side. = Let A be a stictly uppe tiangula n x n atix. Then thee exists an invetible atix P and positive integes n 1,., n k, n 1 n k >0 and n n k = n such that P -1 AP = diag (j n1 (0),.., J nk (0)). Moeove if A has eal enties then P will also have eal enties. (without poof) 8. Jodan decoposition theoe: A C nxn is siila to the atix diag (J n1 ( λ 1 ),. Jn k ( λ k ) ),whee Jni( λ i ) C n i xn i and n 1 + +n k =n. each J ni( λ i )- λ i I ni is of the fo of the above esult. This fo is essentially unique,that its depends only on A and the ode of occuence of eigenvalues. Poof. Only uniqueness is equied to pove now. Fo that note that if λ 0, ank j n = fo all the positive integes n. Fo j (0),ank j (0) n =0,if n and ank j(0) - ank j(0) =1 fo n. wite n(λ)= n n P a g e

5 ank( j ( λ) λi ) n.then n-1 (λ i )- n (λ i ) is the nube of Jodan blocks of size ateleast n appeaing in J.Theefoe the nube of Jodan blocks of size exactly equal to n is ( n-1 (λ i )- n (λ i ))-( n (λ i )- n+1 (λ i ))= n-1 (λ i )-2 n (λ i )+ n+1 (λ i ). Thus two nxn atices A and B ae siila if and only if they have the sae eigen chaacteistics polynoial and fo each eigen value λ and positive intege k, ank (A-λI) k = ank (B-λI) k II. APPLICATIONS IN GROUP THEORY We show that being siila to a Jodan canonical fo atix is a elation that patitions the set of squae atices with coplex enties M n n (C). Of couse because only one equivalence class has the identity atix, Jodan canonical fo does not patition the ing of atices into subings. We study the question of foing an algebaic stuctue fo the equivalence classes ade by patitioning M n (C). We give a lea on patitioning a set into equivalence classes. Lea : Let be an equivalence elation on a set X. Then the collection of equivalence classes C (x) fos a patition of X. So by Lea, if we can show that is an equivalence elation on M n n (C), it will show that patitions that set into subsets all siila to a paticula Jodan canonical fo atix (so the Jodan fo of the subset of each subset epesents the ebes of that subset). Recall fo Canto s diagonal poof that the eal nubes ae uncountable, so the coplex nubes, which contain the eal nubes, ae also uncountable. But note that fo an n n atix A = λi, with λ i C, we can have any eigenvalues fo A in the coplex nubes. Since we have patitioned M n n (C) based on eigenvalues, this gives us uncountably any siilaity equivalence classes. Note that fo siilaity A B expesses A = P 1 BP, fo soe P. But we can have P = I, and then P 1 = I. So we find that is eflexive, because we have an invetible P such that A = P 1 AP, as A = A. If A B, then A = P 1 BP. But then B = P AP 1, and we can just have Q = P 1, which gives us B = Q 1 AQ, so B A, so is syetic. If A B and B C, then A = P 1 BP and B = Q 1 CQ. Note that B = P AP 1, so P AP 1 = Q 1 CQ. Thus A = P 1 Q 1 CQP. Let R = QP. As P and Q ae invetible, R, thei poduct is also invetible. So R 1 = P 1 Q 1. Thus A = R 1 CR, so is tansitive. Thus is an equivalence elation. Hence, by the patitioning lea, patitions M n n (C) into subsets of atices siila to soe atix in Jodan canonical fo (and all siila to each othe). We can show that the poduct of two abitay atices in a cetain equivalence class is not in geneal in that equivalence class; siilaly, the equivalence classes ae not theselves closed unde addition. As well, ultiplication of eleents in these classes is not well defined, as in geneal the poduct of two diffeent choices of atices as epesentatives of thei equivalence class ae not in the sae equivalence class. Fo exaple, let us have the following: A =, B = C = 1587 P a g e

6 Clealy both A and B ae siila to each othe, and B is the Jodan canonical fo epesentative of thei equivalence class. We find that AB = C, and that C is not siila to B 2. Thus we cannot ake any goupoid with atix ultiplication as its opeation out of siilaity classes, because ultiplication of siilaity classes is not well defined. III. CONCLUSION The Jodan canonical fo has any uses in linea and abstact algeba. When it can happen, diagaonalization is quite useful and it is good to know that we can always get soething vey close to diagonal fo with Jodan canonical fo. We also found that we can patition the set of squae atices M n n (C) into equivalence classes of diffeent Jodan canonical fos, but that thee does not see to be any natual algebaic stuctue that can be foed by siilaity classes. REFERENCES [1] R. Fletche and D.C. Soensen. An algoithic deivation of the Jodan canonical fo. Aeican Matheatical Monthly, 90:12-16, [2] A. Galpein and Z. Waksan. An eleentay appoach to Jodan theoy. Aeican Matheatical Monthly,87: , [3] G.H. Golub and J.H. Wilkenson. Ill-conditioned eigensystes and the coputation of the Jodan Canonical Fo. SIAM Review, 18: , [4] K. Hoffan and R. Kunze. Linea Algeba. Pentice-Hall, Englewood Cliffs, New Jesey, [5] H. V aliaho. An eleentay appoach to the Jodan fo of a atix. Aeican Matheatical Monthly, 97: , [6]. Noan J. Bloch, Abstact algeba with applications, Pentice-Hall, Englewood Cliffs, New Jesey, [7] A. J. Colean, The geatest atheatical pape of all tie, Math. Intelligence 11 (1989), no. 3, MR 90f: P a g e