Jordan blocks. Defn. Let λ F, n Z +. The size n Jordan block with e-value λ is the n n upper triangular matrix. J n (λ) =

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1 Jordan blocks Aim lecture: Even over F = C, endomorphisms cannot always be represented by a diagonal matrix. We give Jordan s answer, to what is the best form of the representing matrix. Defn Let λ F, n Z +. The size n Jordan block with e-value λ is the n n upper triangular matrix λ λ 1... J n (λ) = λ λ Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

2 Example E.g. Write down the block diagonal matrix J 2 ( 3) J 3 ( 2) J 1 (7). Defn A matrix is in Jordan canonical form if it is a direct sum of Jordan blocks. Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

3 E-values of Jordan blocks Prop The Jordan block J n (λ 0 ) has λ 0 as its only e-value. The geometric multiplicity is 1 but the algebraic multiplicity is n. Rem When F = C, Jordan blocks are as far from diagonalisable as can be. Proof. J n (λ 0 ) λi is upper triangular with λ 0 λ in all the diagonal entries so cp Jn (λ) = (λ 0 λ) n & the alg mult of λ 0 is n J n (λ 0 ) λ 0 I = which has kernel F e 1 so the geom mult is 1. Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

4 Commutativity of direct sums Prop Let V, W = F-spaces. Then S : V W W V : (v, w) T (w, v) T is an isomorphism. Proof. Indeed, S is linear being given by the matrix ( ) 0 id W id V 0 and it is invertible with inverse ( ) 0 id V. id W 0 Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

5 Permuting diagonal blocks Prop Let A 1,..., A r be square matrices over F & σ S r be a permutation. Then the block diagonal matrices A 1... A r & A σ(1)... A σ(r) are similar. In other words, permuting the diagonal blocks of a matrix does not change its similarity class. Proof. We prove the case r = 2, i.e. A 1 A 2 & A 2 A 1 are similar. The general case can be proved similarly by induction on the r = 2 case or using general permutation matrices. Suppose that A i M ni n i for i = 1, 2. Then ( ) 1 ( ) ( ) 0 In1 A1 0 0 In1 I n2 0 0 A 2 I n2 0 Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

6 Statement of Jordan s canonical form theorem Theorem (Jordan s canonical form) Let F = C (or any alg closed field). 1 Any square matrix A (over F) is similar to a direct sum of Jordan blocks J. Such a J is called a Jordan canonical form for A. 2 If J m1 (λ 1 )... J mr (λ r ) & J n1 (µ 1 )... J ns (µ s ) are similar then r = s & the Jordan blocks are a permutation of each other i.e. there s σ S r such that J ni (µ i ) = J mσ(i) (λ σ(i) ). 3 Hence the set of Jordan blocks with multiplicity, can be considered a similarity invariant. 4 Two matrices are similar iff their Jordan blocks are the same up to permutation. Proof in general will be delayed for several lectures when we have more tools. E.g. The following matrices are not similar. Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

7 Proof of Jordan s theorem for n = 2 Proof n = 2 case. We prove part 1) (existence of Jordan canonical forms), the other parts follow easily from using the similarity invariants of the e-values & their geometric & algebraic multiplicity. Note that the diagonal matrix ( λ λ 2 ) = J1 (λ 1 ) J 2 (λ 2 ) so any diagonalisable matrix is similar to a direct sum of Jordan blocks. Hence the theorem is proved in the case of a diagonalisable matrix. Suppose now that A M 22 (F) is not diagonalisable so by our criterion for diagonalisability, there is an e-value β whose alg mult a is strictly greater than its geom mult m. But 1 m < a 2 so m = 1, a = 2. It suffices to prove in this case the following lemma. Lemma (Suppose A is not diagonalisable). Pick any v 2 F 2 ker(a βi ) & let v 1 = (A βi )v 2. Then the matrix representing A wrt the co-ord system C = (v 1 v 2 ) : F 2 F 2 is J 2 (β). Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

8 Proof lemma Proof. Note dim ker(a βi ) = 1 so we may find v 2 F 2 such that v 2 / ker(a βi ). Also a = 2 so cp A (λ) = (β λ) 2. Note that v 1 is an e-vector since the Cayley-Hamilton thm ensures (A βi )v 1 = (A βi ) 2 v 2 = (βi A) 2 v 2 = 0v 2 = 0 i.e. v 1 ker(a βi ). Now v 2 / ker(a βi ) ensures v 1, v 2 are linearly independent. We show that the matrix B representing A wrt the co-ord system C is J 2 (β). The first column of B is The second column is Hence A is similar to C 1 ACe 1 = C 1 Av 1 = C 1 βv 1 = βe 1. C 1 ACe 2 = C 1 Av 2 = C 1 [(A βi ) + βi ]v 2 = C 1 (v 1 + βv 2 ) = e 1 + βe 2 B = ( ) β 1 = J 0 β 2 (β) which completes the proof of the thm for n = 2. Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

9 Example of computing Jordan canonical form E.g. Let A = ( ). Write A = CJC 1 for some C GL 2 (C) & J a direct sum of Jordan blocks. Daniel Chan (UNSW) Lecture 24: Preview of Jordan canonical forms Semester / 9

Eigenvectors. Prop-Defn

Eigenvectors. Prop-Defn Eigenvectors Aim lecture: The simplest T -invariant subspaces are 1-dim & these give rise to the theory of eigenvectors. To compute these we introduce the similarity invariant, the characteristic polynomial.