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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No , ISSN: (prited versio); ISSN: (o-lie versio) url: doi: PAijpameu ON TENSOR PRODUCT DECOMPOSITION OF k-tridiagonal TOEPLITZ MATRICES Asuka Ohashi 1, Tsuyoshi Sasaki Usuda 1, Tomohiro Sogabe 2, Fatih Yılmaz 3 1 Graduate School of Iformatio Sciece & Techology Aichi Prefectural Uiversity Aichi , JAPAN 2 Graduate School of Egieerig Nagoya Uiversity Nagoya , JAPAN 3 Departmet of Mathematics Polatlı Art ad Sciece Faculty Gazi Uiversity 06900, Akara, TURKEY Abstract: I the preset paper, we provide a decompositio of a k-tridiagoal Toeplitz matrix via tesor product By the decompositio, the required memory of the matrix is reduced ad the matrix is easily aalyzed sice we ca use properties of tesor product AMS Subject Classificatio: 15A18, 15A15 Key Words: k-tridiagoal Toeplitz matrix, decompositio, tesor product, determiat, eigevalues, iteger powers 1 Itroductio A tridiagoal Toeplitz matrix is oe of tridiagoal matrices ad has costat Received: May 14, 2015 Correspodece author c 2015 Academic Publicatios, Ltd url: wwwacadpubleu
2 538 A Ohashi, TS Usuda, T Sogabe, F Yılmaz etries o each diagoal parallel to the mai diagoal It is widely kow that the matrix arises i the fiite differece discretizatio of the differetial equatio (cf [8, 14]) For the recet developmets, see, eg, [1], [2], [4], [11], ad [12] A k-tridiagoal matrix is oe of geeralizatios of a tridiagoal matrix ad has received much attetio i recet years (eg, [3], [6], [9], ad [10]) Here, let T (k) be a -by- k-tridiagoal matrix defied as d a d 2 a 2 0 T (k) := 0 d k a k b (1) k b k+2 d b 0 0 d If d i = d (i = 1,2,,), a i = a (i = 1,2,, k), b i = b (i = k + 1,k + 2,,), where 1 k <, T (k) is a k-tridiagoal Toeplitz matrix Moreover, whe k = 1, T (1) is a ordiary tridiagoal Toeplitz matrix We cosider a k-tridiagoal Toeplitz matrix T (k) Hereafter, tesor product is briefly explaied sice it is used i a decompositio of k-tridiagoal Toeplitz matrices i the preset paper Tesor product is also referred to as the Kroecker product ad represeted by the symbol The defiitio of tesor product of matrices A C m ad B C p q is a 11 B a 12 B a 1 B a 21 B a 22 B a 2 B A B := Cmp q, (2) a m1 B a m2 B a m B where a ij is the (i,j) elemet of A Let â i ad a j be the i-th row ad the j-th colum vectors i A, respectively Similarly, let ˆb i ad b j be the i-th row ad the j-th colum vectors i B, respectively The, A B = â 1 ˆb 1 â 1 ˆb 2 â m ˆb p,
3 ON TENSOR PRODUCT DECOMPOSITION 539 = (a 1 b 1,a 1 b 2,,a b q ), (3) as the other expressios of A B The purpose of the preset paper is to save the required memory of a k- tridiagoal Toeplitz matrix ad to simplify aalyses of that We propose to decompose the k-tridiagoal Toeplitz matrix ito a smaller matrix with the similar structure tha the origial oe ad a idetity matrix via tesor product Eve if the umber i T (k) is very large, the matrix is decomposed ito T (1) 2 ad the idetity matrix uder a certai coditio The, oe eeds oly aalyses of T (1) 2 i order to aalyze T (k) by usig properties of tesor product This paper is orgaized as follows I Sectio 2, we give a theorem of the decompositio via tesor product ad show two examples I Sectio 3, the decompositio is applied i order to simplify the theorem ad the propositio i [13] ad to reduce a computatioal complexity 2 Mai Results I this sectio, we preset a theorem of a decompositio of a k-tridiagoal Toeplitz matrix T (k) ad show two examples First, the theorem is as follows: Theorem 1 Let T (k) be a -by- k-tridiagoal Toeplitz matrix If there exist atural umbers, k, ad m such that = m ad k = mk, where m > 1, T (k) is decomposed ito the form: where I m represets the idetity matrix of order m T (k) = T (k ) I m, (4) Proof First, let T ad S be Toeplitz matrices of the same size The, T is equal to S if ad oly if both of the followig equatios are satisfied: (T) 1: = (S) 1: for the first row vectors i T ad S; (T) :1 = (S) :1 for the first colum vectors i those Here, (T) i: ad (T) :j deote the i-th row ad the j-th colum vectors i T, respectively Sice T (k ) i (4) has Toeplitz structure, T (k ) I m also has Toeplitz structure Hece, the two matrices are the same if both of the first colum ad row vectors i T (k) ad i T (k ) I m are equal From (3), the first row ad colum vectors are obtaied as follows: (T (k ) I m ) 1: = (T (k ) ) 1: (I m ) 1:
4 540 A Ohashi, TS Usuda, T Sogabe, F Yılmaz = (d 1,0,,0,a 1,0,,0) e T 1 = ( d 1 e T 1,0 T m,,0 T m,a 1 e T 1,0 T m,,0 T m), (5) (T (k ) I m ) :1 = (T (k ) ) :1 (I m ) :1 d 1 0 = 0 b k e 1 = d 1 e 1 0 ṃ 0 m b k +1e 1 0 ṃ 0 m, (6) where e 1 ad 0 m represet the m-dimesioal first caoical vector ad the m- dimesioal zerovector, respectively From (5) ad(6), wehave (T (k ) I m ) 1: = (T (k) ) 1: ad (T (k ) I m ) :1 = (T (k) ) :1 Thus, T (k ) I m = T (k) This completes the proof Theorem 1 provides three otes: first, the required memory of T (k) is the lowest i all the values m whe m = gcd(,k); secod, the k -tridiagoal Toeplitz matrix T (k ) is the tridiagoal Toeplitz matrix T (1) uder the coditio that = mk; third, the determiat, the eigevalues, iteger powers, ad the iversio of T (k) are easily computed from those of T (k ) As for the coditio i the secod ote, the origial matrix T (k) is decomposed ito the tridiagoal Toeplitz matrix of order 2 ad the idetity matrix of order k, ie, T (k) = T (1) 2 I k, uder the coditio that = 2k Next, two examples uder the coditio that = mk are show Example 2 Let = 8 ad k = 2 Settig m = 2, the the 2-tridiagoal Toeplitz matrix T (2) 8 is decomposed such as T (2) 8 = T (1) 4 I 2 The matrices are
5 ON TENSOR PRODUCT DECOMPOSITION 541 specifically deoted as follows: T (2) 8 = d 0 a d 0 a b 0 d 0 a b 0 d 0 a b 0 d 0 a b 0 d 0 a b 0 d b 0 d = d a 0 0 b d a 0 0 b d a 0 0 b d I 2 Example 3 Let = 8 ad k = 4 Settig m = 4, the the 4-tridiagoal Toeplitz matrix T (4) 8 is decomposed such as T (4) 8 = T (1) 2 I 4 The matrices are specifically deoted as follows: T (4) 8 = d a d a d a d a b d b d b d b d = ( d a b d ) I 4 From Examples 2 ad 3, we ca cofirm the first ad secod otes Particularly, we ca see that the umber of ozero elemets of matrices is reduced by oe m-th 3 Applicatios I Subsectio 31, we preset some corollaries, which are obtaied from Theorem 1 Some of the corollaries imply the theorem ad the propositio that were proved i [13], however the corollaries i the preset paper have simpler ad more geeral expressios tha i [13] I Subsectio 32, we show that Theorem 1 is used i order to reduce a computatioal complexity
6 542 A Ohashi, TS Usuda, T Sogabe, F Yılmaz 31 The Symmetric k-tridiagoal Toeplitz Matrices We cosider a specialized form of symmetric k-tridiagoal Toeplitz matrices, ie, a, i j = k, (S (k) ) i,j = d, i = j, (7) 0, otherwise, where i,j = 1,2,,, ad parameters ad k are atural umbers such that = mk Applyig Theorem 1 to S (k) i (7), we obtai Corollary 4 Corollary 4 Let S (k) be the matrix as i (7) The S (k) = S (1) m I k Proof By the defiitio of S (k) ad Theorem 1, the result ca be obtaied Usig Corollary 4, the determiat, the eigevalues, ad arbitrary iteger powers of S (k) are easily computed as below Corollary 5 Let S (k) be the matrix as i (7) The det(s (k) ) = [det(s (1) m )] k Proof By Corollary 4 ad the property of the determiat of tesor product, we have det(s (k) ) = det(s(1) m I k) = [det(s (1) m )]k Corollary 6 Let S (k) be the matrix as i (7) The, the eigevalues λ j of S (k) are represeted by ( ) jπ λ j = d+2acos, +1 where j = 1,2,,m Proof The result is obtaied by Corollary 4, the aalytical forms of the eigevalues of the tridiagoal Toeplitz matrix (cf [8, Example 725]), ad the eigevalues of tesor product
7 ON TENSOR PRODUCT DECOMPOSITION 543 Note that the eigevalues are particular forms of those i [7] Corollary 7 Let S (k) be the matrix as i (7) The where r is a arbitrary iteger (S (k) )r = (S (1) m )r I k, Proof By Corollary 4 ad the property of tesor product, the result is obvious Let a = 1, d = 0, ad m = 4 i (7) The Corollary 6 correspods to [13, Propositio 2] As for Corollary 7, we have (S (k) )r = which correspods to [13, Theorem 5] r I k, 32 A Reductio of a Computatioal Complexity by the Decompositio Theorem 1 ca be used i order to reduce the computatioal complexity We here focus o the computatio of the iversio of the k-tridiagoal Toeplitz matrix T (k) usig [5, Theorem 21] A algorithm to compute the iversio with the decompositio is show below First, the k-tridiagoal Toeplitz matrix T (k) is decomposed by Theorem 1 The, the iversio of T (k) is computed by (T (k) ) 1 = (T (k ) ) 1 I m Here, the algorithm i [5, Theorem 21] computes the iversio elemet-wise Therefore, the fewer T (k) has elemets, the lower the computatioal complexity of the algorithm is Sice the umber of ozero elemets of T (k ) is reduced by oe m-th, the computatioal complexity is also reduced by oe m-th 4 Coclusio I the preset paper, we gave a decompositio of a k-tridiagoal Toeplitz matrix via tesor product As applicatios of the decompositio, we have show that the determiat, the eigevalues, ad arbitrary iteger powers of the matrix are
8 544 A Ohashi, TS Usuda, T Sogabe, F Yılmaz easily computed ad that the iversio of the matrix is computed with lower computatioal complexity tha that without the decompositio Ackowledgmets This work has bee supported i part by JSPS KAKENHI(Grat Nos , ) Refereces [1] JW Demmel, Applied Numerical Liear Algebra, SIAM, USA (1997) [2] MEA El-Mikkawy, A geeralized symbolic Thomas algorithm, Appl Math, 3, No 4 (2012), , doi: /am [3] MEA El-Mikkawy, T Sogabe, A ew family of k-fiboacci umbers, Appl Math Comput, 215, No 12 (2010), , doi: /jamc [4] CF Fischer, RA Usmai, Properties of some tridiagoal matrices ad their applicatio to boudary value problems, SIAM, J Numer Aal, 6, No 1 (1969), , doi: / [5] J Jia, T Sogabe, MEA El-Mikkawy, Iversio of k-tridiagoal matrices with Toeplitz structure, Comput Math Appl, 65, No 1 (2013), , doi: /jcamwa [6] E Kilic, O a costat-diagoals matrix, Appl Math Comput, 204, No 1 (2008), doi: /jamc [7] E Kırklar, F Yılmaz, A ote o k-tridiagoal k-toeplitz matrices, Ala J Math, 39, (2015), 1-4 [8] CD Meyer, Matrix Aalysis ad Applied Liear Algebra, SIAM, USA (2004) [9] T Sogabe, MEA El-Mikkawy, Fast block diagoalizatio of k-tridiagoal matrices, Appl Math Comput, 218, No 6 (2011), , doi: /jamc
9 ON TENSOR PRODUCT DECOMPOSITION 545 [10] T Sogabe, F Yılmaz, A ote o a fast breakdow-free algorithm for computig the determiats ad the permaets of k- tridiagoal matrices, Appl Math Comput, 249, (2014), , doi: /jamc [11] J Witteburg, Iverses of tridiagoal Toeplitz ad periodic matrices with applicatios to mechaics, J Appl Math Mech, 62, No 4(1998), , doi: /S (98) [12] T Yamamoto, Iversio formulas for tridiagoal matrices with applicatios to boudary value problems, Numer Fuct Aal Optim, 22, No 3-4 (2001), , doi: /NFA [13] F Yılmaz, T Sogabe, A ote o symmetric k-tridiagoal matrix family adthefiboacciumbers, It J Pure ad Appl Math, 96, No2(2014), , doi: /ijpamv96i210
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