Jordan Normal Form Revisited

Transcription

1 Mathematics & Statistics Auburn University, Alabama, USA Oct 3, 2007 Jordan Normal Form Revisited Speaker: Tin-Yau Tam Graduate Student Seminar Page 1 of 19 tamtiny@auburn.edu

2 Let us start with some online conversations. conversation 1 conversation 2 Page 2 of 19

3 1. Given A, B C n n. A is said to be similar to B if there is a nonsingular matrix P such that P 1 AP = B, denoted by B A. Fact: is an equivalence relation 1. Reflexive (A A): A = IAI = I 1 AI for all A C n n. 2. Symmetric (A B implies A B): P 1 AP = B implies P BP 1 = A. 3. Transitive (A B and B C imply C A) P 1 AP = B and Q 1 BQ = C imply (P Q) 1 A(P Q) = C. Page 3 of 19 As an equivalence relation, similarity partitions C n n classes representative (normal form, canonical form) into equivalence

4 2. Jordan Normal Form Each A C n n is similar to a block diagonal matrix J = J 1... where each block J i is a square matrix of the form (Jordan block) λ i 1. λ.. i J i =. C ni n.. 1 i. Remarks: The eigenvalues λ i and λ k may not be distinct for i k. λ i J p Page 4 of 19

5 Example: A = P 1 AP = J where J = , P = Check: AP = P J. Page 5 of 19

6 A little History: In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Camille Jordan ( ). It appears in the context of a canonical form for linear substitutions over the finite field of order a prime. The Jordan of Gauss-Jordan elimination is Wilhelm Jordan (1842 to 1899). Jordan algebras are called after the German physicist and mathematician Pascual Jordan (1902 to 1980). Page 6 of 19

7 3. Numerical unstable Consider [ ] A = 1 1 ɛ 1. [ ] If ɛ = 0, then the Jordan normal form is simply However, for ɛ 0, the Jordan normal form is [ 1 + ] ɛ ɛ So it is hard to develop a robust numerical algorithm for the Jordan normal form. For this reason, the Jordan normal form is usually avoided in numerical analysis. Matlab command [P, J] = jordan(a) Page 7 of 19

8 4. Remark: Jordan normal form may be useless for numerical linear algebra, it has a valid place in applied linear algebra. For recent development, see Stefano Serra-Capizzano, Jordan canonical form of the Google matrix: a potential contribution to the PageRank computation, SIAM J. Matrix Anal. Appl. 27 (2005), Page 8 of 19 (1) (a) Every A C n n is similar to a complex symmetric matrix. (b) Every A C n n is a product of two complex symmetric matrices; one of which can be chosen nonsingular.

9 (2) A m 0 if and only if the eigenvalue moduli of A are less than 1. λ 1 λ... Reason: It suffices to consider J =. = λi + N C k k... 1 λ Since N m = 0 for all m k, we have m ( ) m m ( ) m J m = (λi + N) m = λ i N m i = λ i N m i i i i=0 i=m k+1 (a) Since the diagonal entries of J m are λ m, if J m 0, then λ m 0, i.e., λ < 1. (b) Conversely, if λ < 1, then ( ) m λ m j = m(m 1)(m 2) (m j + 1)λ m m j j!λ j m j λ m j!λ j 0 as m by l Hopital s rule. It finds application in population growth. Page 9 of 19

10 (3) A is similar to its transpose A T : By Jordan normal form A = P JP 1. So A A T amounts to J J T. Now λ 1 λ = λ 1 λ 1 (4) Solution to the a linear system of ODE: where x (t) = Ax(t), x(0) = x 0 x(t) = P e tj P 1 x 0 = P e tj 1... e tj p 1 t t2 t 2 n 1 (n 1)! t e tj i = e tλ i 1 t n 2 (n 2)!... 1 P 1 x 0, Page 10 of 19

11 5. 1. Brualdi, Richard A., The Jordan canonical form: an old proof. Amer. Math. Monthly 94 (1987), no. 3, Cater, S., An elementary development of the Jordan canonical form. Amer. Math. Monthly 69 (1962), no. 5, Filippov, A. F., of the theorem on reduction of a matrix to Jordan form. Moscow Univ. Bul., no. 2, Fletcher, R.; Sorensen, D. C., An algorithmic derivation of the Jordan canonical form. Amer. Math. Monthly 90 (1983), no. 1, Galperin, A.; Waksman, Z., An elementary approach to Jordan theory. Amer. Math. Monthly 87 (1980), no. 9, Gohberg, Israel; Goldberg, Seymour, A simple proof of the Jordan decomposition theorem for matrices. Amer. Math. Monthly 103 (1996), no. 2, Page 11 of Väliaho, H., An elementary approach to the Jordan form of a matrix. Amer. Math. Monthly 93 (1986), no. 9,

12 6. We switch to the lower triangular version of Jordan normal form: Each A C n n is similar to a block diagonal matrix, i.e., for some nonsingular P P 1 AP = J = J 1... J p where each block J i is a square matrix of the form (Jordan block) J i = λ i 1 λ i λ i C ni n i. An interpretation: P 1 AP is the matrix representation with respect to a new basis given by the columns of P, since P 1 ( )P means a change of basis. Page 12 of 19

13 Partition P = [P 1 P p ] accordingly. Let P i = [v 1 v 2 v ni ] C n ni. Jordan form amounts to AP i = P i J i = P i (λ i I + N i ), N i = From AP i = P i (λ i I + N i ) we have i.e. [Av 1 Av 2 Av ni ] = λ i [v 1 v 2 v ni ] + [v 2 v 3 v ni 0], [(A λ i I)v 1 (A λ i I)v 2 (A λ i I)v ni ] = [v 2 v 3 v ni 0]. In other words, v 1, v 2,..., v ni are related: set v := v 1, m := n i, λ := λ i, Page 13 of 19 v 1 = v, v 2 = (A λi)v,...,..., v m = (A λi) m 1 v (chain)

14 7. A Short proof Roitman, Moshe, of the Jordan decomposition theorem. Linear and Multilinear Algebra 46 (1999), no. 3, In terms of the language of linear operator: Theorem 7.1. Let T : V V be a linear operator acting on a finite dimensional space over C. Then V has a T -Jordan basis, that is, an ordered basis which consists of Jordan sequences: a (T, λ)-jordan sequences, where λ is a scalar, is a sequence of vectors of the form Page 14 of 19 v, (T λi)v,..., (T λi) m 1 v for v V and m 1 such that (T λi) m v = 0.

15 We will use contradiction: (1) Assuming that the theorem is false, let T : V V be a counterexample with dim V 2 minimal, thus V 0. (2) T has an eigenvalue µ in C. Replacing T by T µi we may assume that µ = 0. Thus dim T (V ) < dim V. (3) By the minimality of dim V, T (V ) has a T 0 -Jordan basis, where T 0 is the restriction of T to T (V ), i.e., T 0 = T T (V ) : T (V ) T (V ), T 0 (w) = T (w) (4) Let V be a subspace of maximal dimension among all subspaces of V (a) invariant under T, and (b) with a Jordan basis, B, containing a basis of T (V ). Clearly T (V ) V V. Page 15 of 19

16 (5) Claim: T (V ) = T (V ). Clearly T (V ) T (V ) since V V. To prove T (V ) T (V ) we will show B T (V ) T (V ). Indeed if w B T (V ), then w belongs to a (T, λ)-jordan sequence u, (T λi)u,..., (T λi) m 1 u B V. Case 1: λ = 0. If w u (not the first one), then w = T k u T (V ), 1 k m. If w = u, pick any w V such that T (w ) = w since w B T (V ) T (V ). If w V, then we may add w to the above sequences thus extending B to a Jordan basis of a subspace properly containing V, a contradiction. Hence w V, i.e., w T (V ). Case 2: λ 0. Observation: for v V v T (V ) (T λi)v = T v λv T (V ). Since w B V and (T λi) m w = 0 T (V ) we see that (T λi) m 1 w T (V ), (T λi) m 2 w T (V ),, w T (V ). Page 16 of 19 Since B contains a basis of T (V ), we have T (V ) = T (V ).

17 Next let v V. Thus for some v V, so that v v ker T. T v = T v We have ker T V, otherwise we may add to B a nonzero vector in ker T, to obtain a Jordan basis of a subspace properly containing V. Thus v V so that V = V, a contradiction. Page 17 of 19

18 Remarks: (1) The proof works for any algebraically closed field. (2) The above proof leads to the construction of a Jordan basis: Example: T is nilpotent of index m, i.e., T m = 0 and T m 1 0. (1) Start with a basis of T m 1 (V ). (2) For each chosen element v we pick an element v in T m 2 (V ) such that T (v ) = v. (3) By adding elements in ker T T m 2 (V ) (thus start a new Jordan sequence) we obtain a Jordan basis of T m 2 (V ), etc. Page 18 of 19

19 T HAN K YOU FOR YOUR AT T EN T ION Page 19 of 19