A Note on Spectra of Optimal and Superoptimal Preconditioned Matrices

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1 A Note on Spectra of Optimal and Superoptimal Preconditioned Matrices Che-ManCHENG Xiao-QingJIN Seak-WengVONG WeiWANG March 2006 Revised, November 2006 Abstract In this short note, it is proved that given any positive definite Hermitian matrix, the eigenvalues of the superoptimal preconditioned matrix do not exceed the corresponding eigenvalues of the optimal preconditioned matrix. Key Words: optimal preconditioner, superoptimal preconditioner, spectrum, preconditioned matrix. AMS subject classifications: 65F10, 65F15. In 1988, T. Chan[2] proposed a circulant preconditioner for Toeplitz systems. For any Toeplitz matrix T n, T. Chan s circulant preconditioner c F (T n ) is defined to be the minimizer of T n W n F where F is the Frobenius norm and W n runs over all circulant matrices. Thec F (T n )iscalledtheoptimalcirculantpreconditioner. In1992,Tyrtyshnikov [6] proposed another circulant preconditioner t F (T n ) which is defined to be the minimizer of I n Wn 1 T n F where W n runs over all nonsingular circulant matrices. Recently this preconditioner (also called superoptimal preconditioner) has been found to be a good approximation in signal processing when there exists a noise[4]. Since the optimal and superoptimal preconditioners are defined not just for Toeplitz matricesbutforgeneralmatricesaswell,webeginwiththegeneralcase. Givenaunitary DepartmentofMathematics,UniversityofMacau,Macao,China. fstcmc@umac.mo. DepartmentofMathematics,UniversityofMacau,Macao,China. xqjin@umac.mo. Theresearch of this author is supported by the Grant 050/2005/A from FDCT. DepartmentofMathematics,UniversityofMacau,Macao,China. swvong@umac.mo. DepartmentofMathematics,UniversityofMacau,Macao,China. 1

2 matrixu C n n,let M U {U Λ n U Λ n isanyn ndiagonalmatrix}. (1) Wenotethatin(1),whenU =F,theFouriermatrix,M F isthesetofallcirculantmatrices [3]. Let δ(e n ) denote the diagonal matrix whose diagonal is equal to the diagonal of the matrixe n. Wehavethefollowinglemma. Lemma 1 [1]ForanyA n C n n,letc U (A n )betheminimizerof W n A n F overallw n M U. Thenc U (A n )isuniquelydeterminedbya n andisgivenby c U (A n ) U δ(ua n U )U. WhenA n ishermitianpositivedefinite,thenc U (A n )isalsohermitianpositivedefinite. Let t U (A n )betheminimizerof I n W 1 n A n F over all nonsingular W n M U. If both A n and c U (A n ) are invertible, then t U (A n ) is uniquelydeterminedbya n andisgivenby t U (A n )=c U (A n A n )[c U(A n )] 1. IfA n ishermitianpositivedefinite,wethenhavebylemma1, [c U (A n )] 1 A n = U [δ(ua n U )] 1 UA n [δ(ua n U )] 1 2(UA n U )[δ(ua n U )] 1 2 where means issimilarto,and [t U (A n )] 1 A n = c U (A n )[c U (A 2 n)] 1 A n M, (2) = U δ(ua n U )[δ(ua 2 n U )] 1 UA n δ(ua n U )[δ(ua 2 nu )] 1 UA n U (δ(ua n U )[δ(ua 2 nu )] 1 ) 1 2UA n U (δ(ua n U )[δ(ua 2 nu )] 1 ) 1 2 N. (3) 2

3 Wethereforehaveby(2)and(3), N = [δ(ua 2 nu )] 1 2 [δ(uan U )] 1 2 (UAn U ) [δ(ua n U )] 1 2 [δ(ua 2 n U )] 1 2 = [δ(ua 2 n U )] 1 2δ(UA n U ) M δ(ua n U )[δ(ua 2 n U )] 1 2 = DMD, (4) where D [δ(ua 2 n U )] 1 2δ(UA n U )=diag(d 1,...,d n ). Nowletusshowthefollowinglemmafirst. Lemma 2 ForanymatrixA n C n n,wehave δ(ua n A nu ) δ(ua n U ) δ(ua nu ) 0. Proof: Foreachk=1,...,n,wehave [δ(ua n A n U )] kk = [δ(ua n U UA n U )] kk = n (UA n U ) kt (UA n U ) tk = t=1 n (UA n U ) kt (UA n U ) kt t=1 (UA n U ) kk (UA n U ) kk =[δ(ua n U )] kk [δ(ua nu )] kk 0. Thus, δ(ua n A nu ) δ(ua n U ) δ(ua nu ) 0. We immediately have the following result. Corollary 1 LetA n beahermitianpositivedefinitematrix. Thenthediagonalentriesof satisfy0<d i 1fori=1,...,n. D [δ(ua 2 nu )] 1 2 δ(uan U )=diag(d 1,...,d n ) We have the following theorem about the spectra of optimal and superoptimal preconditioned matrices by using Corollary 1 and Courant-Fischer Theorem[5]. 3

4 Theorem 1 LetA n beahermitianpositivedefinite matrix. If the eigenvaluesareordered in the following decreasing way, λ 1 λ 2 λ n, thenwehave fork=1,2,...,n. λ k ([t U (A n )] 1 A n ) λ k ([c U (A n )] 1 A n ), Proof: Wehaveby(2)-(4),Courant-FischerTheoremandCorollary1thatfork=1,...,n, λ k ([t U (A n )] 1 A n ) = λ k (N)=λ k (DMD) = min max x DMDx y 1,...,y k 1 C n x 1, x y 1,...,y k 1 = min max (Dx) M(Dx) y 1,...,y k 1 C n x 1, (Dx) D 1 y 1,...,D 1 y k 1 min max (Dx) M(Dx) y 1,...,y k 1 C n Dx 1, (Dx) D 1 y 1,...,D 1 y k 1 z=dx y 1,...,y k 1 C n z 1, z D 1 y 1,...,D 1 y k 1 D 1 y 1,...,D 1 y k 1 C n z 1, z D 1 y 1,...,D 1 y k 1 u i =D 1 y i u 1,...,u k 1 C n z 1, z u 1,...,u k 1 = λ k (M)=λ k ([c U (A n )] 1 A n ). Hereweremark that{y 1,...,y k 1 }runsthroughallpossiblechoicesofk 1vectorsinC n ifandonlyif{d 1 y 1,...,D 1 y k 1 }runsthroughallpossiblechoicesofk 1vectorsinC n. Also,theouterminimizationisomittedwhenk=1. Remark. LetD=diag(d 1,...,d n )begivenasincorollary1. Suppose α=max{d 1,...,d n } 1. Since x 1implies Dx α,onecaneasilymodifytheproofoftheorem1tohave λ k ([t U (A n )] 1 A n ) α 2 λ k ([c U (A n )] 1 A n ), k=1,...,n. 4

5 This α may of theoretical interests for further analysis. However, the computation of its valueisnoteasyingeneral. References [1] R. Chan, X. Jin and M. Yeung, The Circulant Operator in the Banach Algebra of Matrices, Linear Algebra Appl., Vol. 149(1991), pp [2] T. Chan, An Optimal Circulant Preconditioner for Toeplitz Systems, SIAM J. Sci. Statist. Comput., Vol. 9(1988), pp [3] P. Davis, Circulant Matrices, 2nd edition, Chelsea Publishing, New York, [4] C. Estatico and S. Serra-Capizzano, The Superoptimal Approximation: the Case of Unbounded Symbols, submitted. [5] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, [6] E. Tyrtyshnikov, Optimal and Superoptimal Circulant Preconditioners, SIAM J. Matrix Anal. Appl., 13(1992), pp