On Some Inverse Singular Value Problems with Toeplitz-Related Structure

Transcription

1 O Some Iverse Sigular Value Problems with Toeplitz-Related Structure Zheg-Jia Bai Xiao-Qig Ji Seak-Weg Vog Abstract I this paper, we cosider some iverse sigular value problems for Toeplitz-related matrices We costruct a Toeplitz-plus-Hakel matrix from prescribed sigular values icludig a zero sigular value The we fid a solutio to the iverse sigular value problem for Toeplitz matrices which have double sigular values icludig a double zero sigular value Keywords Iverse sigular value problem, Toeplitz matrix, Toeplitz-plus-Hakel matrix AMS subject classificatio 65F15, 65F18 1 Itroductio The problem of recostructig a matrix from give eigevalues has applicatios i may differet areas icludig applied mechaics, geophysics, cotrol desig, pricipal compoet aalysis, particle physics, ad circuit theory, etc Mathematically, this problem is called a iverse eigevalue problem (IEP) For details of mathematical theory ad algorithmic aspects of the IEP, readers ca refer to 1,, 3, 5, 8, 9, 10, 1, 16, 19 I this paper, we cosider the iverse sigular value problem (ISVP) which is a atural geeralizatio of the IEP The mai cocer of a ISVP is the recostructio of a matrix with sigular values matchig some prescribed values This problem has practical applicatios i optimal sequece desig for direct-spread code divisio multiple access 17 We ca refer to 7 for early work o the ISVP Recetly, the iterest o the study of the ISVP has bee growig I 14, 15, the ISVP for oegative ad positive matrices has bee cosidered Backward error aalysis for the ISVP 7 is studied i 6 Very recetly, a Ulm-like method is proposed by the authors for solvig a kid of ISVP 18 The cocer of this paper is o the study of structured ISVP Our iterest lies o Toeplitzrelated structure I 4, solutios to the IEP for Toeplitz-plus-Hakel matrices have bee School of Mathematical Scieces, Xiame Uiversity, Xiame , P R Chia (zjbai@xmueduc) The research of this author was partially supported by the Natural Sciece Foudatio of Fujia Provice of Chia for Distiguished Youg Scholars (No 010J0600) ad NCET Departmet of Mathematics, Uiversity of Macau, Macao, P R Chia (xqji@umacmo) The research of this author was supported by the grat UL00/08-Y3/MAT/JXQ01/FST from Uiversity of Macau Departmet of Mathematics, Uiversity of Macau, Macao, P R Chia (swvog@umacmo) The research of this author was supported by the grat MYRG06(Y1-L1)-FST11-VSW from Uiversity of Macau 1

2 studied Diele et al i 004 costructed solutios to the Toeplitz-plus-Hakel IEP i a close form 11 The mai igrediets of their costructio are some idempotet rak-oe Toeplitz-plus- Hakel matrices which are obtaied by cosiderig the properties of the Chebyshev polyomials Ispired by their result, we fid that, i a similar fashio, solutios to the ISVP with Toeplitzplus-Hakel matrices ca also be obtaied i a close form With this beig established, as i 11, we costruct a solutio to the ISVP for Toeplitz matrices which have double sigular values icludig a double zero sigular value This paper is orgaized as follows I Sectio, based o some properties of the discrete cosie trasform matrix ad the discrete sie trasform matrix, we costruct a Toeplitz-plus- Hakel matrix from prescribed sigular values icludig a zero sigular value I Sectio 3 we provide a solutio to the ISVP for Toeplitz matrices from give double sigular values icludig a double zero sigular value ISVP for Toeplitz-plus-Hakel matrices We first recall the discrete cosie trasform matrix ad the discrete sie trasform matrix Let W be the -by- discrete cosie trasform matrix with etries ( ) δj1 (i 1)(j 1)π W ij = cos, 1 i, j, where δ ij is the Kroecker delta; see for istace 13, p150 Let U be the -by- discrete sie trasform matrix with etries 11 ( ) δj (i 1)jπ U ij = si, 1 i, j We kow that both W ad U are orthogoal Defie V := w,, w, w 1, where w j deotes the jth colum of W By usig the orthogoal matrices U ad V, we ca costruct a collectio of -by- rak-oe Toeplitz-plus-Hakel matrices Propositio 1 For k = 1,, 1, let u k ad v k deote the kth colums of U ad V, respectively The the -by- rak-oe matrices A k = u k vk T, for k = 1,, 1, are Toeplitzplus-Hakel Proof: Let α k = with k = 1,, 1 Notice that si α cos β = 1 si(α + β) + si(α β), for all α, β R Thus the (i, j)th etry of A k is give by ( k(i 1)π A k ij = αk si ( = α k k(i j)π si ) cos ) + si ( ) k(j 1)π ( k(i + j 1)π )

3 Let ad T k := α k H k := α k = α k 0 si kπ si k( )π si k( 1)π si kπ 0 si kπ si k( )π 0 si kπ si k( )π si kπ 0 si k( )π si k( 1)π si kπ si kπ si kπ si k( 1)π si kπ si kπ si k(+1)π si k( 1)π si kπ si kπ si kπ si kπ si k( )π si k(+1)π si k( )π si k( 1)π si kπ si k( 1)π 0 0 si k( 1)π si k( 1)π 0 si kπ 0 si k( 1)π si kπ si kπ (1) () The we have A k = T k + H k, where T k is Toeplitz ad H k is Hakel, for k = 1,, 1 Similarly, we ca also fid a collectio of -by- rak-oe Hakel-mius-Toeplitz matrices Propositio For k = 1,, 1, let u k ad v k deote the kth colums of U ad V, respectively The the -by- rak-oe matrices B k = v k u T k, for k = 1,, 1, are Hakelmius-Toeplitz Proof: Let α k = with k = 1,, 1 Notice that cos α si β = 1 si(α + β) si(α β), for all α, β R Thus the (i, j)th etry of B k is give by ( ) k(i 1)π B k ij = αk cos si ( k(i + j 1)π = α k si 3 ( ) k(j 1)π ) ( k(i j)π si )

4 Let T k ad H k be defied as i (1) ad (), respectively The we obtai B k = H k T k Corollary 3 The matrices T k ad H k defied i (1) ad () are skew-cetrosymmetric, ie, for k = 1,, 1, where is called the -by- ati-idetity matrix T k J = JT k, H k J = JH k, J := O the ISVP for Toeplitz-plus-Hakel matrices, we have the followig propositio Propositio 4 Give σ 1 σ σ = 0, the matrix is Toeplitz-plus-Hakel Moreover, A = 1 σ k A k = σ k A k A = UΣV T, Σ = diag(σ 1,, σ ), ie, {σ k } are the sigular value of A ad v k ad u k are the left-sigular ad right-sigular vectors for σ k, respectively Proof: Sice A k is Toeplitz-plus-Hakel for k = 1,, 1, we kow that A is Toeplitz-plus- Hakel Moreover, 1 1 A = σ k A k = σ k u k vk T = UΣV T, where {u k } ad {v k} are orthogoal colums of U ad V, respectively By usig the similar argumets of Propositio 4, we ca derive the followig result o the ISVP for Hakel-mius-Toeplitz matrices Propositio 5 Give σ 1 σ σ = 0, the matrix 1 B = σ k B k = σ k B k 4

5 is Hakel-mius-Toeplitz Moreover, B = V ΣU T, Σ = diag(σ 1,, σ ), ie, {σ k } are the sigular value of B ad u k ad v k are the left-sigular ad right-sigular vectors for σ k, respectively 3 Toeplitz ISVP with double sigular values I this sectio, we costruct a Toeplitz matrix from give double sigular values We focus o the followig Toeplitz ISVP: Give σ 1 σ σ = 0, fid a -by- Toeplitz matrix T such that {σ k } are double sigular values of T O the solvability of the Toeplitz ISVP, we have the followig result Theorem 31 Give σ 1 σ σ = 0, let T ad H be defied by 1 1 T := σ k T k, H := σ k H k, (3) where T k ad H k are defied as i (1) ad () The the matrix T JH T := HJ T (4) is a -by- Toeplitz matrix such that {σ k } are double sigular values of T Proof: By the defiitios of T k ad H k, it is easy to check that the matrices Tk JH k T k :=, k = 1,, 1, H k J are all -by- Toeplitz matrices Hece, T = 1 O the other had, we have by Propositios 1,, 4, ad 5, T k A = T + H, B = H T T k is a -by- Toeplitz matrix Thus, ad 1 JV T V 1 JU T U = 1 JU U = 1 JV V Σ, Σ, 5

6 where U, V, ad Σ are defied as i Sectio Therefore, T = U Σ V T, where U = 1 JU U JV V Σ 0, Σ = 0 Σ, V = 1 JV V JU U By Corollary 3, T k ad H k are skew-cetrosymmetric The T ad H defied i (3) are also skew-cetrosymmetric Therefore, the Toeplitz matrix T defied i (4) is skew-cetrosymmetric Refereces 1 Z J Bai, Iexact Newto methods for iverse eigevalue problems, Appl Math Comput, Vo 17 (006), pp Z J Bai, R H Cha, ad B Morii, A iexact Cayley trasform method for iverse eigevalue problems, Iverse Problems, Vol 0 (004), pp Z J Bai ad X Q Ji, A ote o the Ulm-like method for iverse eigevalue problems, Recet Advaces i Scietific Computig ad Matrix Aalysis, pp 1 7 Eds: X Q Ji, H W Su, ad S W Vog, Iteratioal Press of Bosto, Bosto; ad Higher Educatio Press, Beijig, D Bii ad M Capovai, Spectral ad computatioal properties of bad symmetric Toeplitz matrices, Liear Algebra Appl Vol 5/53 (1983), pp R H Cha, H L Chug, ad S F Xu, The iexact Newto-like method for iverse eigevalue problem, BIT, Vol 43 (003), pp X S Che, A backward error for the iverse sigular value problem, J Comput Appl Math, Vol 34 (010), pp M T Chu, Numerical methods for iverse sigular value problems, SIAM J Numer Aal, Vol 9 (199), pp M T Chu, Iverse eigevalue problems, SIAM Rev, Vol 40 (1998), pp M T Chu ad G H Golub, Structured iverse eigevalue problems, Acta Numer, Vol 11 (00), pp M T Chu ad G H Golub, Iverse Eigevalue Problems: Theory, Algorithms, ad Applicatios, Oxford Uiversity Press, Oxford, 005 6

7 11 F Diele, T Laudadio, ad N Mastroardi, O some iverse eigevalue problems with Toeplitz-related structure, SIAM J Matrix Aal Appl, Vol 6 (004), pp S Friedlad, J Nocedal, ad M L Overto, The formulatio ad aalysis of umerical methods for iverse eigevalue problems, SIAM J Numer Aal, Vol 4 (1987), pp A Jai, Fudametals of Digital Image Processig, Pretice Hall, Eglewood Cliffs, NJ, E Motaño, M Salas, ad R L Soto, Noegative matrices with prescribed extremal sigular values, Comput Math Appl, Vol 56 (008), pp E Motaño, M Salas, ad R L Soto, Positive matrices with prescribed sigular values, Proyeccioes, Vol 7 (008), pp W P She, C Li, ad X Q Ji, A Ulm-like method for iverse eigevalue problems, Appl Numer Math, Vol 61 (011), pp J A Tropp, I S Dhillo, ad R W Heath Jr, Fiite-step algorithms for costructig optimal CDMA sigature sequeces, IEEE Tras Iform Theory, Vol 50 (004), pp S W Vog, Z J Bai, ad X Q Ji, A Ulm-like method for iverse sigular value problems, SIAM J Matrix Aal Appl, to appear 19 S F Xu, A Itroductio to Iverse Eigevalue Problems, Pekig Uiversity Press ad Vieweg Publishig, Beijig,