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jordan_decomp

# jordan_decomp - Jordan Decomposition Theorem LinearAlg...

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Jordan Decomposition Theorem : LinearAlg Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.uﬂ.edu/ squash/ 7 November, 2010 (at 18:32 ) Abstract: Gives a home-grown proof of the Jor- dan Decomposition Theorem. ( Some of the lemmas work in Hilbert space. ) The “Partial-form JCF Theorem”, (26), needs to be reworked. Prolegomenon Our goal is to prove the JCF ( Jordan Canonical Form ) Theorem for a linear trn T : H H , where H is a ﬁnite-dim’al vectorspace. Formally, we’ll assume that H is F ×H , where the ﬁeld F is ei- ther R or C . For vectorspaces use vectorspace: H , A , B , E , V dimension: H , A , B , E , V . Use sans-serif font for matrices A , B , G , I , M . For square matrices A e , let Diag ( A 1 ,..., A E ) be the partitioned matrix which has A 1 A E along its diagonal, and zeros elsewhere. 1 : Notation . A collection C := { V 1 , V 2 V L } of subspaces of H is linearly independent ( abbreviation lin-indep ) if the only soln to v 1 + ··· + v L = 0 , with each v V , is the trivial soln v 1 = 0 v L = 0 . Recall that a subspace V H is T -invariant if T ( V ) V . I’ll use eval , evec and e-space for eigenvalue , eigenvector and eigenspace . ± § 1 Examining nilpotent case In sections § 1 and § 2, “eval” means the eigen- value zero , and “evec” means an eigenvector with eval zero . 2 : Defn . W.r.t T , a vector v is nilpotent if T d ( v ) = 0 for some posint d . Indeed, the T - depth of a vector v , written T -Depth( v ), is the inﬁmum of all natnums n for which T n ( v ) is 0 . The zero-vector has depth 0. An evec for eval=0 has depth 1. ( A non-nilpotent vector has depth . ) Use Nil( T ) for the nilspace of T ; it comprises the set of ﬁnite-depth vectors. So Nil( T ) def == [ n =1 Ker( T n ) note Ker( T ) . Transformation T is nilpotent if there exists a posint D such that T D = 0 . Since H is ﬁnite di- mensional, trn ± ² ³ ´ T is nilpotent iﬀ Nil( T ) = H . ± 3 : Depth Lemma ( preliminary ). Consider a sum v 1 + v 2 + + v L 3 0 : whose depths satisfy d 1 > d 2 > ... > d L . Then the depth of (3 0 ) is d 1 . Proof. Exercise. A downtup ( “down tuple” ) -→ D = ( D 1 ,...,D E ) is a sequence of integers with D 1 > D 2 > ... > D E > 1 . 4 : The size of D is the sum D 1 + + D E . A posint D determines a D × D Jordan Block ma- trix JB ( D ) := 0 1 0 1 0 1 . . . . . . 0 1 0 5 : with zeros on the diagonal and ones on the ﬁrst oﬀ-diagonal. Every undisplayed position is zero. 6 : Nilpotent JCF Theorem. A nilpotent T : F ² has a unique downtup D so that M = M ( D ) := Diag ± ( D 1 ) ( D E ) ² 7 : is the matrix of T w.r.t some basis. In particular, Size( D ) equals H . Webpage http://www.math.uﬂ.edu/ squash/ Page 1 of 7

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Page 2 of 7 The Construction Prof. JLF King Remark. In general, the above basis is not unique. The theorem can be restated ITOf matrices. A nilpotent F -matrix M 0 determines a unique downtup -→ D so that, with M from (7) , M 0 = G 1 · M ( D ) · G , for some invertible F -matrix G . ± Temporarily letting c 1 ,..., c D denote the stan- dard basis, notice that the D × D jordan-blk (5) acts on the standard basis by sending c D → ··· → c 1 0 . Let this motivate our deﬁnition of a chain : a sequence C = ( c d ) D d =1
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jordan_decomp - Jordan Decomposition Theorem LinearAlg...

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