Iterative method for computing a Schur form of symplectic matrix

Transcription

1 Aals of the Uiversity of Craiova, Mathematics ad Computer Sciece Series Volume 421, 2015, Pages ISSN: Iterative method for computig a Schur form of symplectic matrix A Mesbahi, AH Betbib, A Kaber, ad F Ellaggoue Abstract We preset i this paper a costructive iterative method to compute a orthogoal ad symplectic Schur form This structured Schur form reductio is used to compute eigevalues ad ivariat subspaces of symplectic matrices Key words ad phrases Symplectic matrix, Symplectic reflector, ortho-sr decompositio, Schur decompositio 1 Itroductio The computatio of the eigevalues ad eigevectors of matrices play importat roles i may applicatios i the physical scieces For example, they play a promiet role i image processig applicatios Measuremet of image sharpess ca be doe usig the cocept of eigevalues [6] I this paper we are iterestig i a structured eigevalue problem We propose a practical method to compute eigevalues ad ivariat subspaces for symplectic matrices witch are of particular importace i applicatios such as optimal cotrol [7, 8] Symplectic matrices are widely used i SR factorizatio witch is a pricipal step i structure-preservig methods It is a logstadig ope problem to compute the eigevalues ad the Schur form i particular the oe how preserved the structure [3, 4] I the case of a eigevalue problem of structured matrices, the preservatio of this structure may help exploit the symmetry of the problem ad to improve the accuracy ad efficiecy calculatios of ivariat subspaces ad eigevalues [5] Our proposed method here is based o a orthogoal ad symplectic SR factorizatio based o orthogoal ad symplectic reflectors Orthogoality is used to preserve the stability ad Symplecticity is used to preserve the structure I paragraph 3, we give ad prove a orthogoal ad symplectic Schur form theorem for real symplectic matrix A algorithm that compute this structured Schur form reductio is give ad umerical experimetal results are preseted to illustrate the effectiveess of our approach 2 Termiology, otatio ad some basic facts I this sectio we collect several well-kow properties of structured matrices, for future referece A ubiquitous matrix i this work is the skew-symmetric matrix O I J 2, where I I O ad O are the idetity ad zero matrix, respectively Note that J2 1 J 2 T J 2 I the followig, we will drop the This paper has bee preseted at Cogrès MOCASIM, Marrakech, November

2 ITERATIVE METHOD FOR COMPUTING A SCHUR FORM OF SYMPLECTIC MATRIX 159 subscript ad 2 wheever the dimesio of correspodig matrix is clear from its cotext The J-traspose of ay 2-by-2p matrix M is defied by M J J T 2pM T J 2 R 2p 2 Ay matrix S R 2 2p satisfyig the property S T J 2 S J 2p S J S I 2p is called symplectic matrix This property is also called J-orthogoality The symplectic similarity trasformatios preserve structured matrix Remark 21 A augmeted matrix S is symplectic if ad oly if the matrix P I P 11 0 P I 0 0 P 21 0 P 22 P11 P 12 is symplectic P 21 P Symplectic reflectors Settig E i [e i e +i ] R 2 2 for i 1,, where e i is the i-th elemet of the caoical basis of R 2 We the obtai Ei J Ei T ad Ei J E j δ ij I 2 { 1 if i j, where δ ij 0 if i j We recall the symplectic reflector o R 2 2 see [1, 2], which is defied i parallel with elemetary reflectors Propositio 21 Let U, V be two 2-by-2 real matrices satisfyig U J U V J V I 2 If the 2-by-2 matrix C I 2 +V J U is osigular, the, the trasformatio S U + V C 1 U +V J I 2 is symplectic ad takes U to V It s called a symplectic reflector Additioally, if U J U T ad V J V T, the S is orthogoal ad symplectic Propositio 22 Let u R 2 be a ozero 2-compoet real vector The reflector S U + αe 1 αi 2 + αe J 1 U 1 U + αe 1 J I 2 where U [u Ju], is orthogoal ad symplectic ad verify Su αe 1 where α u T u u 2 2 Proof Sice U J U αi 2 with α u T u u 2 2 > 0, the a simple calculatio gives the result 22 Symplectic matrices Propositio 23 If M R 2 2 is symplectic the M T is symplectic Proof Sice MJ M T MJM T JMM 1 J 1 MJJM 1 J 1 I J 1 J M T JM J MJ M T MJM T JMM 1 J 1 MJJM 1 J 1 I J 1 J This proves that M T is symplectic

3 160 A MESBAHI, AH BENTBIB, A KANBER, AND F ELLAGGOUNE A C Propositio 24 If M R B D 2 2 where A, B, C, D R, the M is symplectic if ad oly if the matrices A, B, C, D verify : { A T B ad C T D are symmetric, A T D B T C I Proof We have The M T B JM T A A T B B T C A T D D T A C T B D T C C T D M is symplectic { B T A A T B ad D T C C T D A T D B T C D T A C T B I { A T B ad C T D are symmetric, A T D B T C I Theorem 25 A matrix M R 2 2 is orthogoal ad symplectic if ad oly if M is i the followig form A B M B A where A, B R verify { A T B is symmetric, A T A + B T B I A C Proof If M is orthogoal ad symplectic, where A, B, C, D R B D, the J MJM T MJM 1 That proves JM MJ The C B ad D A By the result of Propositio 24, A, B R verify { A T B is symmetric, A T A + B T B I 23 Ortho-SR decompositio By usig ortho-symplectic reflectors defied above, we decompose a symplectic real 2-by-2 matrix A o the form A SR where S R 2 2 is orthogoal ad symplectic ad R is symplectic ad is i the followig form 0 R

4 ITERATIVE METHOD FOR COMPUTING A SCHUR FORM OF SYMPLECTIC MATRIX 161 Algorithm : Ortho-symplectic SR-decompositio Hereafter, Matlab otatios are used Iput : A R 2 2 a symplectic matrix Output: A 2 2 orthogoal ad symplectic matrix S ad a symplectic matrix R i the form 1 such that A SR For k1 to do u : A:, k; E k [e k, Je k ]; I eye2; I k blkdiag eyek, eye k, eyek, eye k For j1 to k-1 do uj 0; u + j 0 ed For If u 0 the u u u U u, J u C I 2Ek J U If det C 0 the T U + EC 1 U + E k J I k ed If ed For else C I + 2E J U T U + EC 1 U + E k J + I ed If S : ST ; A : T A 3 Ortho-symplectic Schur form Lemma 31 If A is a symplectic matrix ad u C 2 a eigevector correspodig to a eigevalue λ of A Suppose that λ is outside the uit circle, the there exists a uitary ad symplectic reflector S such that Su αe 1 Proof Let u be a eigevector correspodig to a eigevalue λ of A such that λ 1 ad u 1 U [u, Ju] C 2 2 is uitary ad symplectic U H U U J U I 2 Ideed, U J U J T U H JU u H Ju u H u u H u u H Ju The matrix A is symplectic, the A H JA J ad, u H Ju 1 λ 2 AuH JAu 1 λ 2 uh Ju u 2 u H Ju u H Ju u 2 Sice λ 1, the u H Ju 0 Let set N E J 1 U If 1 is ot eigevalue of the 2-by-2 matrix N, the the reflector S U + E 1 I 2 + E J 1 U 1 U + E 1 J I 2 is

5 162 A MESBAHI, AH BENTBIB, A KANBER, AND F ELLAGGOUNE uitary ad symplectic ad verify SU αe 1 The Su αe 1 here α 1 If 1 is eigevalue of N, the 1 is ot eigevalue of E1 J U Ideed, we have E1 J U e T 1 u e T 1 Ju u1 u e T +1u e T +1 the dete +1Ju u +1 u 1 J U u u The 1 reflector S U E 1 I 2 E1 J U 1 U E 1 J I 2 is uitary ad symplectic ad verify SU αe 1 ad the Su αe 1 here α 1 Theorem 32 Let A be a 2-by-2 real symplectic matrix such that all eigevalues of A are outside the uit circle The there exists a uitary ad symplectic matrix Q such that A QRQ T where R is symplectic ad is i the followig form R Proof The proof ca be see by usig the pricipe of iductio o Let λ be a eigevalue of A ad u a ormalized correspodig eigevector U [u, Ju] R 2 2 is uitary ad symplectic U H U U J U I From Lemma 31 there exists a uitary ad symplectic reflector S 1 such that S 1 u αe 1 ad the S 1 AS T 1 e 1 λe 1 The matrix R 1 S 1 AS1 H is symplectic ad it is i the followig form λ 0 S 1 AS1 H R λ 0 0 λ vt 0 A C x w T X B y 0 Y D

6 ITERATIVE METHOD FOR COMPUTING A SCHUR FORM OF SYMPLECTIC MATRIX 163 A B The matrix R 2 is also symplectic Ideed, C D A B R 1 C D, A, B, C, D R We have R T C 1 JR 1 T A A T C C T B A T D D T A B T C D T B B T D Ad R T C 2 JR 2 T A A T C C T B A T D D T A B T C D T B B T D By block multiplicatio, we have C T A A T C 0 O 0 C T A A T C, C T B A T D 1 0 X C T B A T D D T B B T D x X X D T B A T D, D T A B T C 1 X X D T A B T C R 1 is symplectic the R 1 T JR 1 J ad Cosequetly C T A A T C 0, C T B A T D I, D T A B T C I ad D T B B T D 0 C T A A T C 0, C T B A T D I, D T A B T C I ad D T B B T D 0 That proves that R 2 is symplectic see, Propositio 24 By iductio hypothesis there exist uitary ad symplectic reflector, S 2, such that S 2 R 2 S2 T R2, R

7 164 A MESBAHI, AH BENTBIB, A KANBER, AND F ELLAGGOUNE Let S 2 1 s 1 1,1 s 1 1, 1 s 1 2,1 s 1 2, 1 1 s 1 1,1 s 1 1, 1 1 s 1,1 s 1, 1 s 1 +1,1 s 1 +1, 1 1 s 1 2 1,1 s 1 2 1, 1 1 s 1 1, s 1 1,2 1 s 1 2, s 1 2,2 1 s 1 1, s 1 1,2 1 s 1, s 1,2 1 s 1 +1, s 1 +1,2 1 s 1 2 1, s 1 2 1,2 1 We take S s 1 1,1 s 1 1, 1 0 s 1 2,1 s 1 2, 1 0 s 1 1,1 s 1 1, s 1,1 s 1, 1 0 s 1 +1,1 s 1 +1, 1 0 s 1 2 1,1 s 1 2 1, s 1 1, s 1 1,2 1 0 s 1 2, s 1 2,2 1 0 s 1 1, s 1 1, s 1, s 1,2 1 0 s 1 +1, s 1 +1,2 1 0 s 1 2 1, s 1 2 1,2 1 The S 2 is uitary ad symplectic ad verify S 2 S 1 AS2 H S1 H R were R is symplectic ad i the followig form 0 R Algorithm 31 Algorithm : Iterative method to compute Schur form of symplectic matrix

8 ITERATIVE METHOD FOR COMPUTING A SCHUR FORM OF SYMPLECTIC MATRIX 165 Iput : A R 2 2 Symplectic, X 0 R 2 Output: Ortho-symplectic Schur form of A V X 0 Repeat W AV W SR ortho-symplectic SR decompositio V S[:, 1 : ] RR R[1 :, 1 : ] util Covergece 32 Numerical examples We compared ad tested the umerical results obtaied by Algorithm 31 with Matlab eig fuctio Our umerical experimets were carried out with Matlab R2009a ad ru it o a Core Duo Petium processor The symplectic matrix A is obtaied from the matrix M M 0 M T where M diagv ad by usig symplectic similarity trasformatios radomly geerated by symplectic reflectors as : M M A S 0 M T S J Example 31 I this example 20 ad v is costructed as follows For k 1 to fix/2 Here Matlab otatio is used vk k + 1 ad v 2 + k k 1

9 166 A MESBAHI, AH BENTBIB, A KANBER, AND F ELLAGGOUNE Example 32 I this example 10 ad v is costructed as follows For k 1 to fix/2 Here Matlab otatio is used vk 1 k+1 ad v 2 + k k Refereces [1] S Agoujil ad A H Betbib, O the reductio of Hamilotoia matrices to a Hamiltoia Jorda caoical form, It Jour Math Stat IJMS , [2] S Agoujil ad A H Betbib, New symplectic trasformatio o C 2 2 : Symplectic reflectors, It Jour of Tomography ad Statistics IJTS , [3] P Beer, V Mehrma, ad H Xu, A ew method for computig the stable ivariat subspace of a real Hamiltoia matrix, J Comput Appl Math , o 1, [4] P Beer, V Mehrma, ad H Xu, A umerically stable, structure preservig method for computig the eigevalues of real Hamiltoia or symplectic pecils, Numer Math , [5] M Dosso ad M Sadkae, A spectral trichotomy method for symplectic matrices, Numer Algor , [6] H Gaidhae, V Hote, ad VSigh, A New Approach for Estimatio of Eigevalues of Images, IJCA , o 9, 1 6 [7] I Gohberg, P Lacaster, ad L Rodma, Idefiite Liear Algebra ad Applicatios, Birkhauser Verlag, Basel, 2005 [8] P Lacaster ad L Rodma, Algebraic Riccati Equatios, Claredo Press, Oxford, 1995 A Mesbahi, F Ellaggoue Departmet of Mathematics ad Iformatics, Uiversity of Souk Ahras, Algeria address: amerbrahim@gmailcom AH Betbib, A Kaber Departmet of Mathematics, Cadi Ayyad Uiversity, Faculty of Sciece ad Techology, Marrakech 42000, Morocco address: abetbib@ucama, kaber@ucama