e AT where is called a Jordan block of size with eigenvalue so 1 Jordan canonical form 122 brPage 3br is upper bidiagonal diagonal is the special case of Jordan blocks of size 1 Jordan form is unique up to permutations of the blocks can have multipl ID: 25207 Download Pdf

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e AT where is called a Jordan block of size with eigenvalue so 1 Jordan canonical form 122 brPage 3br is upper bidiagonal diagonal is the special case of Jordan blocks of size 1 Jordan form is unique up to permutations of the blocks can have multipl

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EE263 Autumn 2007-08 Stephen Boyd Lecture 12 Jordan canonical form Jordan canonical form generalized modes Cayley-Hamilton theorem 121

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Jordan canonical form what if cannot be diagonalized? any matrix can be put in Jordan canonical form by a similarity transformation, i.e. AT where is called a Jordan block of size with eigenvalue (so =1 Jordan canonical form 122

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is upper bidiagonal diagonal is the special case of Jordan blocks of size = 1 Jordan form is unique (up to permutations of the blocks) can have multiple blocks with same eigenvalue

Jordan canonical form 123

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note: JCF is a conceptual tool , never used in numerical computations! ) = det( sI ) = ( hence distinct eigenvalues = 1 diagonalizable dim λI is the number of Jordan blocks with eigenvalue more generally, dim λI min k, n so from dim λI for = 1 ,. .. we can determine the sizes of the Jordan blocks associated with Jordan canonical form 124

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factor out and λI λI for, say, a block of size 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 = 0 for other blocks (say, size 3, for 1) 2)( 0 ( 0 0 ( Jordan canonical form 125

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Generalized eigenvectors suppose AT diag , ... ,J express as = [ where are the columns of associated with th Jordan block we have AT let = [ in then we have: Av i.e. , the ﬁrst column of each is an eigenvector associated with e.v. for = 2 , ... ,n Av ij i j ij the vectors , ... v in are sometimes called generalized eigenvectors Jordan canonical form 126

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Jordan form LDS consider LDS Ax by change of coordinates , can put into form system is decomposed into independent Jordan block systems /s /s /s Jordan blocks are sometimes called Jordan chains (block diagram

shows why) Jordan canonical form 127

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Resolvent, exponential of Jordan block resolvent of Jordan block with eigenvalue sI +1 = ( + ( + ( where is the matrix with ones on the th upper diagonal Jordan canonical form 128

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by inverse Laplace transform, exponential is: tJ t tF + ( 1)!) t 1)! 2)! Jordan blocks yield: repeated poles in resolvent terms of form t in tA Jordan canonical form 129

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Generalized modes consider Ax , with (0) = in then ) = Te Jt (0) = trajectory stays in span of generalized eigenvectors coeﬃcients have form λt ,

where is polynomial such solutions are called generalized modes of the system Jordan canonical form 1210

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with general (0) we can write ) = tA (0) = Te tJ (0) = =1 tJ (0)) where hence: all solutions of Ax are linear combinations of (generalized) modes Jordan canonical form 1211

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Cayley-Hamilton theorem if ) = is a polynomial and , we deﬁne ) = Cayley-Hamilton theorem: for any we have ) = 0 , where ) = det( sI example: with 1 2 3 4 we have ) = , so ) = 7 10 15 22 1 2 3 4 = 0 Jordan canonical form 1212

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corollary: for every , we have

span I, A, A , ... , A (and if is invertible, also for i.e. , every power of can be expressed as linear combination of I, A,. .., A proof: divide into to get ) + is remainder polynomial then ) + ) = ) = Jordan canonical form 1213

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for : rewrite C-H theorem ) = = 0 as /a /a (1 /a is invertible = 0 ) so /a /a (1 /a i.e. , inverse is linear combination of = 0 ,.. ., n Jordan canonical form 1214

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Proof of C-H theorem ﬁrst assume is diagonalizable: AT = ) = ( since ) = ) = (Λ) it suﬃces to show (Λ) = 0 (Λ) = ( ( diag (0 , ,. .., diag

,. .. , 0) = 0 Jordan canonical form 1215

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now lets do general case: AT ) = ( suﬃces to show ) = 0 ) = ( 0 1 0 0 0 1 {z = 0 Jordan canonical form 1216

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