Jordan Normal Form Revisited Home Page Title Page

Home , Canonical form

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

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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

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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 PBP −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

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

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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

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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 ... µ = 0. Thus Applications Proofs dim T (V ) < dim V. Basis change A short proof (3) By the minimality of dim V , T (V ) has a T0-Jordan basis, where T0 is the restriction of T to T (V ), i.e., Home Page

Title Page T0 = T |T (V ) : T (V ) → T (V ),T0(w) = T (w) JJ II

J I (4) Let V 0 be a subspace of maximal dimension among all subspaces of V Page 15 of 19 (a) invariant under T , and Go Back (b) with a Jordan basis, B, containing a basis of T (V ). Full Screen Clearly T (V ) ⊂ V 0 ⊂ V . Close Quit (5) Claim: T (V ) = T (V 0). Clearly T (V 0) ⊂ T (V ) since V 0 ⊂ V . To prove T (V ) ⊂ T (V 0) we will show B ∩ T (V ) ⊂ T (V 0). Indeed if w ∈ B ∩ T (V ), then w belongs to a (T, λ)-Jordan sequence

m−1 0 Similarity u, (T − λI)u, . . . , (T − λI) u ∈ B ⊂ V . Jordan Form Numerical ... Case 1: λ = 0. If w =6 u (not the ﬁrst one), then w = T ku ∈ T (V 0), Applications Proofs 0 0 1 ≤ k ≤ m. If w = u, pick any w ∈ V such that T (w ) = w since w ∈ Basis change A short proof B ∩ T (V ) ⊂ T (V ). If w0 6∈ V 0, then we may add w0 to the above sequences 0 thus extending B to a Jordan basis of a subspace properly containing V , a Home Page 0 0 0 contradiction. Hence w ∈ V , i.e., w ∈ T (V ). Title Page 0 Case 2: λ =6 0. Observation: for v ∈ V JJ II

J I v ∈ T (V 0) ⇔ (T − λI)v = T v − λv ∈ T (V 0). Page 16 of 19

Since w ∈ B ⊂ V 0 and (T − λI)mw = 0 ∈ T (V 0) we see that Go Back

Full Screen m−1 0 m−2 0 0 (T − λI) w ∈ T (V ), (T − λI) w ∈ T (V ), ··· , w ∈ T (V ). Close

Since B contains a basis of T (V ), we have T (V ) = T (V 0). Quit Similarity 0 0 Next let v ∈ V . Thus for some v ∈ V , Jordan Form Numerical ... T v = T v0 Applications Proofs Basis change so that v − v0 ∈ ker T . A short proof

Home Page We have ker T ⊂ V 0, otherwise we may add to B a nonzero vector in ker T , Title Page to obtain a Jordan basis of a subspace properly containing V 0. JJ II

J I Thus v ∈ V 0 so that V 0 = V , a contradiction. Page 17 of 19

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Quit Similarity Jordan Form Remarks: (1) The proof works for any algebraically closed ﬁeld. Numerical ... (2) The above proof leads to the construction of a Jordan basis: Applications Proofs Basis change A short proof Example: T is nilpotent of index m, i.e., T m = 0 and T m−1 =6 0. m−1 (1) Start with a basis of T (V ). Home Page 0 m−2 (2) For each chosen element v we pick an element v in T (V ) such that Title Page 0 T (v ) = v. JJ II m−2 (3) By adding elements in ker T ∩T (V ) (thus start a new Jordan sequence) J I m−2 we obtain a Jordan basis of T (V ), etc. Page 18 of 19

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Quit Similarity Jordan Form Numerical ... T HAN K YOU FOR Applications Proofs Basis change YOURATTENTION A short proof

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