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Jordan Normal Form Revisited Home Page Title Page Mathematics & Statistics Auburn University, Alabama, USA Similarity Jordan Form Numerical ... Applications Proofs Basis change Oct 3, 2007 A short proof Jordan Normal Form Revisited Home Page Title Page JJ II Speaker: Tin-Yau Tam J I Page 1 of 19 Graduate Student Seminar Go Back Full Screen [email protected] Close Quit Similarity Jordan Form Numerical ... Applications Proofs Basis change Let us start with some online conversations. A short proof conversation 1 Home Page conversation 2 Title Page JJ II J I Page 2 of 19 Go Back Full Screen Close Quit 1. Similarity Similarity Jordan Form Numerical ... Given A, B ∈ n×n. A is said to be similar to B if there is a nonsingular C Applications matrix P such that P −1AP = B, denoted by B ∼ A. Proofs Basis change Fact: Similarity is an equivalence relation A short proof 1. Reflexive (A ∼ A): A = IAI = I−1AI for all A ∈ . Cn×n Home Page 2. Symmetric (A ∼ B implies A ∼ B): P −1AP = B implies P BP −1 = Title Page A. JJ II J I 3. Transitive (A ∼ B and B ∼ C imply C ∼ A) P −1AP = B and Page 3 of 19 Q−1BQ = C imply (PQ)−1A(PQ) = C. Go Back As an equivalence relation, similarity partitions Cn×n into equivalence Full Screen classes → representative (normal form, canonical form) Close Quit 2. Jordan Normal Form Similarity Each A ∈ Cn×n is similar to a block diagonal matrix Jordan Form Numerical ... J1 Applications . Proofs J = .. Basis change A short proof Jp Home Page where each block Ji is a square matrix of the form (Jordan block) Title Page λ 1 i JJ II .. λ . J I J = i ∈ . i . Cni×ni .. 1 Page 4 of 19 Go Back λi Full Screen Remarks: The eigenvalues λi and λk may not be distinct for i =6 k. Close Quit Example: Similarity 5 4 2 1 Jordan Form Numerical ... 0 1 −1 −1 Applications A = . −1 −1 3 0 Proofs Basis change 1 1 −1 2 A short proof P −1AP = J where Home Page 1 0 0 0 −1 1 1 1 Title Page 0 2 0 0 1 −1 0 0 JJ II J = ,P = . 0 0 4 1 0 0 −1 0 J I 0 0 0 4 0 1 1 0 Page 5 of 19 Check: AP = PJ. Go Back Full Screen Close Quit Similarity A little History: Jordan Form Numerical ... In 1870 the Jordan canonical form appeared in Treatise on substitutions and Applications algebraic equations by Camille Jordan (1838-1922). It appears in the context Proofs Basis change of a canonical form for linear substitutions over the finite field of order a A short proof prime. Home Page Title Page The Jordan of Gauss-Jordan elimination is Wilhelm Jordan (1842 to 1899). JJ II J I Jordan algebras are called after the German physicist and mathematician Pas- Page 6 of 19 cual Jordan (1902 to 1980). Go Back Full Screen Close Quit 3. Numerical unstable Similarity Consider Jordan Form " # Numerical ... 1 1 A = . Applications 1 Proofs Basis change " # A short proof 1 1 If = 0, then the Jordan normal form is simply . However, for =6 0, 0 1 Home Page the Jordan normal form is Title Page " √ # 1 + 0 √ . JJ II 0 1 − J I So it is hard to develop a robust numerical algorithm for the Jordan normal Page 7 of 19 Go Back form. For this reason, the Jordan normal form is usually avoided in numerical Full Screen analysis. Close Matlab command [P, J] = jordan(A) Quit 4. Applications Similarity Remark: Jordan normal form may be useless for numerical linear algebra, it Jordan Form has a valid place in applied linear algebra. Numerical ... Applications Proofs Basis change For recent development, see A short proof Stefano Serra-Capizzano, Jordan canonical form of the Google matrix: a potential contribution to the PageRank computation, SIAM J. Matrix Anal. Home Page Title Page Appl. 27 (2005), 305–312. JJ II J I Page 8 of 19 Go Back (1) (a) Every A ∈ Cn×n is similar to a complex symmetric matrix. Full Screen (b) Every A ∈ n×n is a product of two complex symmetric matrices; one C Close of which can be chosen nonsingular. Quit (2) Am → 0 if and only if the eigenvalue moduli of A are less than 1. λ 1 .. λ . Similarity Reason: It suffices to consider J = = λI + N ∈ Ck×k. Jordan Form .. . 1 Numerical ... Applications λ Proofs m Since N = 0 for all m ≥ k, we have Basis change A short proof m m X m X m J m = (λI + N)m = λiN m−i = λiN m−i i i Home Page i=0 i=m−k+1 Title Page (a) Since the diagonal entries of J m are λm, if J m → 0, then λm → 0, i.e., JJ II |λ| < 1. J I (b) Conversely, if |λ| < 1, then Page 9 of 19 m j m m m−j m(m − 1)(m − 2) ··· (m − j + 1)λ m λ Go Back λ = j ≤ j → 0 m − j j!λ j!λ Full Screen as m → ∞ by l’Hopital’s rule. Close It finds application in population growth. Quit (3) A is similar to its transpose AT : By Jordan normal form A = PJP −1. So A ∼ AT amounts to J ∼ J T . Now λ 1 λ 1 1 . . . . 1 .. = .. .. 1 .. 1 λ 1 λ 1 Similarity Jordan Form Numerical ... Applications (4) Solution to the a linear system of ODE: Proofs Basis change A short proof 0 x (t) = Ax(t), x(0) = x0 Home Page Title Page etJ1 JJ II tJ −1 . −1 x(t) = P e P x0 = P .. P x0, J I tJp e Page 10 of 19 where Go Back t2 tn−1 1 t 2 ··· (n−1)! Full Screen tn−2 1 t ··· Close tJi tλi (n−2)! e = e . .. Quit 1 5. Proofs 1. Brualdi, Richard A., The Jordan canonical form: an old proof. Amer. Math. Monthly 94 (1987), no. 3, 257–267. Similarity 2. Cater, S., An elementary development of the Jordan canonical form. Amer. Jordan Form Math. Monthly 69 (1962), no. 5, 391–393. Numerical ... Applications 3. Filippov, A. F., A short proof of the theorem on reduction of a matrix to Jordan Proofs Basis change form. Moscow Univ. Bul., 26 1971 no. 2, 18–19. A short proof 4. Fletcher, R.; Sorensen, D. C., An algorithmic derivation of the Jordan canoni- Home Page cal form. Amer. Math. Monthly 90 (1983), no. 1, 12–16. Title Page 5. Galperin, A.; Waksman, Z., An elementary approach to Jordan theory. Amer. JJ II Math. Monthly 87 (1980), no. 9, 728–732. J I 6. Gohberg, Israel; Goldberg, Seymour, A simple proof of the Jordan decom- Page 11 of 19 position theorem for matrices. Amer. Math. Monthly 103 (1996), no. 2, Go Back 157–159. Full Screen 7. Valiaho,¨ H., An elementary approach to the Jordan form of a matrix. Amer. Close Math. Monthly 93 (1986), no. 9, 711–714. Quit 8. ............... 6. Basis change We switch to the lower triangular version of Jordan normal form: Each A ∈ Similarity Cn×n is similar to a block diagonal matrix, i.e., for some nonsingular P Jordan Form Numerical ... Applications J1 Proofs −1 . Basis change P AP = J = .. A short proof Jp Home Page where each block Ji is a square matrix of the form (Jordan block) Title Page λi JJ II 1 λi J I Ji = ∈ n ×n . . C i i Page 12 of 19 .. .. Go Back 1 λi Full Screen −1 An interpretation: P AP is the matrix representation with respect to a new Close −1 basis given by the columns of P , since P (·)P means a change of basis. Quit Partition P = [P1 | · · · | Pp] accordingly. Let Pi = [v1 | v2 | · · · | vni] ∈ Cn×ni. Jordan form amounts to 0 Similarity Jordan Form 1 0 Numerical ... AP = P J = P (λ I + N ), N = i i i i i i i .. .. Applications . Proofs 1 0 Basis change A short proof From APi = Pi(λiI + Ni) we have Home Page Title Page [Av1 | Av2 | · · · | Avni] = λi[v1 | v2 | · · · | vni] + [v2 | v3 | · · · vni | 0], JJ II i.e. J I Page 13 of 19 [(A − λiI)v1 | (A − λiI)v2 | · · · | (A − λiI)vni] = [v2 | v3 | · · · | vni | 0]. Go Back In other words, v1, v2, . , vni are related: set v := v1, m := ni, λ := λi, Full Screen Close m−1 v1 = v, v2 = (A − λI)v, . , . , vm = (A − λI) v (chain) Quit 7. A Short proof Similarity Jordan Form Roitman, Moshe, A short proof of the Jordan decomposition theorem. Numerical ... Applications Linear and Multilinear Algebra 46 (1999), no. 3, 245–247. Proofs Basis change A short proof In terms of the language of linear operator: Home Page Theorem 7.1. Let T : V → V be a linear operator acting on a finite dimen- Title Page sional space over . Then V has a T -Jordan basis, that is, an ordered basis C JJ II which consists of Jordan sequences: a (T, λ)-Jordan sequences, where λ is a J I scalar, is a sequence of vectors of the form Page 14 of 19 Go Back v, (T − λI)v, . , (T − λI)m−1v Full Screen for v ∈ V and m ≥ 1 such that (T − λI)mv = 0. Close Quit 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 =6 0. Similarity (2) T has an eigenvalue µ in . Replacing T by T − µI we may assume that Jordan Form C Numerical ..
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