Fast Algorithms for Solving RFPrLR Circulant Linear Systems

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1 Fast Algorithms for Solving RFPrLR Circulant Linear Systems ZHAOLIN JIANG Department of Mathematics Linyi University Shuangling Road Linyi city CHINA JING WANG Department of Mathematics Linyi University Shuangling Road Linyi city CHINA Abstract: In this paper fast algorithms for solving RFPrLR circulant linear systems are presented by the fast algorithm for computing polynomials The unique solution is obtained when the RFPrLR circulant matrix over the complex field C is nonsingular the special solution general solution are obtained when the RFPrLR circulant matrix over the complex field C is singular The extended algorithms is used to solve the RLPrFL circulant linear systems Examples show the effectiveness of the algorithms Key Words: RFPrLR circulant matrix Linear system Fast algorithm 1 Introduction Circulant matrix family have important applications in various disciplines including image processing [1] communications [2] signal processing [3] encoding [4] preconditioner [5] They have been put on firm basis with the work of P Davis [6] Z L Jiang [7] The circulant matrices long a fruitful subject of research [ ] have in recent years been extended in many directions [ ] The fx-circulant matrices are another natural extension of this well-studied class can be found in [ ] The fx-circulant matrix has a wide application especially on the generalized cyclic codes [13] The properties structures of the x n rx 1- circulant matrices which are called RFPrLR circulant matrices are better than those of the general fx- circulant matrices so there are good algorithms for solving the RFPrLR circulant linear system In this paper the fast algorithms presented avoid the problems of error efficiency produced by computing a great number of triangular functions by means of other general fast algorithms There is only error of approximation when the fast algorithm is realized by computers only the elements in the first row of the RFPrLR circulant matrix the constant term r are used by the fast algorithm so the result of the computation is accurate in theory Specially the result computed by a computer is accurate over the rational field Definition 1 A row first-plus-rlast right RFPrLR circulant matrix with the first row a 0 a 1 a is meant a square matrix over the complex field C of the form A RFPrLRcircfra 0 a 1 a a 0 a 1 a a a 0 + ra a n 2 1 a 2 a 3 + ra 2 a 1 a 1 a 2 + ra 1 a 0 + ra It can be seen that the matrix over the complex field C with an arbitrary first row the following rule for obtaining any other row from the previous one: Get the i+1st row by adding r times the last element of the ith row to the first element of the ith row then shifting the elements of the ith row cyclically one position to the right Note that the RFPrLR circulant matrix is a x n rx 1 circulant matrix [13] when r 0 A becomes a circulant matrix [6 7] when r 1 A becomes a RFPLR circulant matrix[ ] when r 1 A becomes a RFMLR circulant matrix [ ] Thus it is the extension of circulant matrix RFPLR circulant matrix RFMLR circulant matrix In this paper let r be a complex number satisfy r n n n It is easily verified that the polynominal gx x n rx 1 has no repeated roots in 1 n the complex field if r n n n 1 n We define Θ 1r as the basic RFPrLR circulant matrix that is E-ISSN: Issue 11 Volume 12 November 13

2 Θ 1r r 0 0 n n 2 It is easily verified that the polynomial gx x n rx 1 is both the minimal polynomial the characteristic polynomial of the matrix Θ 1r ie Θ 1r is nonsingular nonderogatory In addition Θ n 1r I n + rθ 1r In view of the structure of the powers of the basic RFPrLR circulant matrix Θ 1r it is clear that A RFPrLRcircfra 0 a 1 a a i Θ i 1r 3 i0 i0 Thus A is a RFPrLR circulant matrix if only if A fθ 1r for some polynomial fx The polynomial fx a i x i will be called the representer of the RFPrLR circulant matrix A In addition the product of two RFPrLR circulant matrices is a RFPrLR circulant matrix the inverse of a nonsingular RFPrLR circulant matrix is also a RFPrLR circulant matrix Furthermore a RFPrLR circulant matrices commute under multiplication Definition 2 A row last-plus-rfirst left RLPrFL circulant matrix with the first row a 0 a 1 a is meant a square matrix over the complex field C of the form A RLPrFLcircfra 0 a 1 a a 0 a 1 a a 1 ra 0 + a a 0 a n 2 ra n 3 + a n 4 a n 3 ra 0 + a ra n 2 + a n 3 a n 2 4 It can be seen that the matrix over the complex field C with an arbitrary first row the following rule for obtaining any other row from the previous one: Get the i+1st row by adding r times the first element of the ith row to the last element of the ith row then shifting the elements of the ith row cyclically one position to the left Obviously the RLPrFL circulant matrix over the complex field C is the extension of left circulant matrix[6 7 36] For the convenience of application we give the obvious results in following lemmas Lemma 3 Let A RFPrLRcircfra 0 a 1 a be a RFPrLR circulant matrix over C let B RLPrFLcircfra a n 2 a 1 a 0 be a RLPrFL circulant matrix over C Then or Î n BÎn A B AÎn Lemma 4 [16] Let C[x] be the polynomial ring over a field C let fx gx C[x] Suppose that the polynomial matrix fx 1 0 gx 0 1 is changed dx ux vx 0 sx tx by a series of elementary row operations then fx gx dx fxux + gxvx dx Lemma 5 C[x]/ x n rx 1 C[Θ 1r ] Proof Consider the following C-algebra homomorphism φ : C[x] C[Θ 1r ] fx A fθ 1r for fx C[x] It is clear that φ is an C-algebra epimorphism So we have C[x]/kerφ C[Θ 1r ] Since C[x] is a principal ideal integral domain there is a monic polynomial px C[x] such that kerφ px Since x n rx 1 is the minimal polynomial of Θ 1r then px x n rx 1 E-ISSN: Issue 11 Volume 12 November 13

3 By Lemma 5 we have the following lemma Lemma 6 Let A RFPrLRcircfra 0 a 1 a be a RFPrLR-circulant matrix Then A is nonsingular if only if fx gx 1 fx a i x i is the representer of A i0 gx x n rx 1 Proof A is nonsingular if only if fx + x n rx 1 is an invertible element in C[x]/ x n rx 1 By Lemma 5 if only if there exists ux + x n rx 1 C[x]/ x n rx 1 such that fxux + x n rx x n rx 1 if only if there exist ux vx C[x] such that fxux + x n rx 1vx 1 if only if fx x n rx 1 1 In [37] for an m n matrix A any solution to the equation AXA A is called a generalized inverse of A In addition if X satisfies X XAX then A X are said to be semi-inverses A {12} In this paper we only consider square matrices A In [38] the smallest positive integer k for which ranka k+1 ranka k holds is called the index of A If A has index 1 the generalized inverse X of A is called the group inverse A # of A Clearly A X are group inverses if only if they are semi-inverses AX XA In [39] [] a semi-inverse X of A was considered in which the nonzero eigenvalues of X are the reciprocals of the nonzero eigenvalue of A These matrices were called spectral inverses It was shown in [] that a nonzero matrix A has a unique spectral inverse A s if only if A has index 1: when A s is the group inverse A # of A 2 Fast algorithms for solving the RFPrLR circulant linear system the RLPrFL circulant linear system Consider the RFPrLR circulant linear system AX b 6 A is given in Equation 1 X x 1 x 2 x n T b b b 1 b 0 T If A is nonsingular then Equation 6 has a unique solution X A 1 b If A is singular Equation 6 has a solution then the general solution of Equation 6 is X A {12} b + I n A {12} AY Y is an arbitrary n-dimension column vector The key of the problem is how to find A 1 b A {12} b A {12} A For this purpose we at first prove the following results Theorem 7 Let A RFPrLRcircfra 0 a 1 a be a nonsingular RFPrLR circulant matrix of order n over C b b b 1 b 0 T Then there exists a unique RFPrLR circulant matrix C RFPrLRcircfrc 0 c 1 c of order n over C such that the unique solution of Equation 6 is the last column of C ie X c c 1 c 0 + rc T i0 Proof Since matrix A RFPrLRcircfra 0 a 1 a is nonsingular then by Lemma 6 we have fx gx 1 fx a i x i is the representer of A gx x n rx 1 Let B RFPrLRcircfrb 0 rb b 1 b be the RFPrLR circulant matrix of order n constructed by b b b 1 b 0 T Then the representer of B is bx b 0 rb + b i x i i1 Therefore we can change the polynomial matrix fx 1 0 bx gx ux vx 0 sx tx cx c 1 x by a series of elementary row operations By Lemma 4 we have ux vx sx tx fx gx 1 0 E-ISSN: Issue 11 Volume 12 November 13

4 ie ux vx sx tx bx 0 cx c 1 x fxux + gxvx 1 uxbx cx Substituting x by Θ 1r in the above two equations respectively we have Since then fθ 1r uθ 1r + gθ 1r vθ 1r I n uθ 1r bθ 1r cθ 1r fθ 1r A gθ 1r 0 bθ 1r B AuΘ 1r I n 7 uθ 1r B cθ 1r 8 By Equation 7 we know that uθ 1r is a unique inverse A 1 of A According to Equation 8 the characters of the RFPrLR circulant matrix we know that the last column of C is c c 1 c 0 + rc T A 1 b Since AA 1 b b then A 1 b c c 1 c 0 + rc T is the solution of Equation 6 Since both A 1 B are unique then A 1 B is also unique So is unique X c c 1 c 0 + rc T A 1 b Theorem 8 Let A RFPrLRcircfra 0 a 1 a be a singular RFPrLR circulant matrix of order n over C b b b 1 b 0 T If the solution of Equation 6 exists then there exist a unique RFPrLR circulant matrix C RFPrLRcircfrc 0 c 1 c a unique RFPrLR circulant matrix E RFPrLRcircfre 0 e 1 e of order n over C such that X 1 c c 1 c 0 + rc T is a unique special solution of Equation 6 X 2 X 1 + I n EY is a general solution of Equation 6 Y is an arbitrary n-dimension column vector Proof Since A RFPrLRcircfra 0 a 1 a then the representer of A is fx a i x i the characteristic polynomial of Θ 1r is i0 gx x n rx 1 fx We can change the polynomial matrix gx dx by a series of 0 elementary row operations Since A is singular by Lemma 4 Lemma 6 we know that dx is the largest common factor which is not equal to 1 of fx gx Let then fx dxf 1 x gx dxg 1 x f 1 x g 1 x 1 Since dx g 1 x 1 we have fx g 1 x dxf 1 x g 1 x 1 Since dx g 1 x 1 fx g 1 x 1 we have fxdx g 1 x 1 Let B RFPrLRcircfrb 0 rb b 1 b be the RFPrLR circulant matrix of order n constructed by b b b 1 b 0 T Then the representer of B is bx b 0 rb + b i x i i1 Therefore we can change the polynomial matrix fxdx 1 0 dxbx dxfx g 1 x ux vx cx 0 sx tx c 1 x ex e 1 x by a series of elementary row operations Then by Lemma 4 we have fxdxux + g 1 xvx 1 9 dxuxbx cx 10 E-ISSN: Issue 11 Volume 12 November 13

5 Therefore dxuxfx ex 11 fxdxuxfx + gxf 1 xvx fx 12 Substituting x by Θ 1r in Equations respectively we have dθ 1r uθ 1r bθ 1r cθ 1r dθ 1r uθ 1r fθ 1r eθ 1r fθ 1r dθ 1r uθ 1r fθ 1r + gθ 1r f 1 Θ 1r vθ 1r fθ 1r Since fθ 1r A gθ 1r 0 bθ 1r B then dθ 1r uθ 1r B cθ 1r 13 dθ 1r uθ 1r A eθ 1r 14 AdΘ 1r uθ 1r A A 15 In the same way we have dθ 1r uθ 1r AdΘ 1r uθ 1r dθ 1r uθ 1r 16 By Equation we know T dθ 1r uθ 1r is a semi-inverses A {12} of A Let let C T B cθ 1r RFPrLRcircfrc 0 c 1 c E T A eθ 1r RFPrLRcircfre 0 e 1 e According to Equation 13 the characters of the RFPrLR circulant matrix we know that the last column of C is c c 1 c 0 + rc T T b Since AX b has a solution then AT b b ie T b is a solution of Equation 6 We have A[T b + I n EY ] AT b + AI n EY b + AY AEY b + AY AT AY b + AY AY b so X 2 X 1 + I n EY is the general solution of Equation 6 Y is an arbitrary n-dimension column vector X 1 T b Since both A T are RFPrLR circulant matrices then AT T A If there exists another RFPrLR circulant matrix T 1 such that AT 1 A A T 1 AT 1 T 1 T 1 A AT 1 Let AT T A H AT 1 T 1 A F Clearly H 2 H F 2 F Thus we have So H F Hence H AT AT 1 AT F H F T 1 A T 1 AT A F H T T AT HT F T T 1 AT T 1 H T 1 F T 1 AT 1 T 1 So T is unique Hence T B C T A E T b are also unique By Theorem 7 Theorem 8 we have the following fast algorithm for solving the RFPrLR circulant linear system 6: Step 1 From the RFPrLR circulant linear system 6 we get the polynomial fx a i x i gx x n rx 1 i0 bx b 0 rb + b i x i ; i1 Step 2 Change the polynomial matrix fx bx gx 0 dx cx by a series of elementary row 0 c 1 x operations; Step 3 If dx 1 then the RFPrLR circulant linear system 6 has a unique solution Substituting x by Θ 1r in polynomial cx we obtain a RFPrLR circulant matrix C cθ 1r RFPrLRcircfrc 0 c 1 c So the unique solution of AX b is c c 1 c 0 + rc T ; Step 4 If dx 1 dx dividing gx we get the quotient g 1 x change the polynomial matrix fxdx dxbx dxfx dxfxbx g 1 x E-ISSN: Issue 11 Volume 12 November 13

6 1 cx ex rx 0 c 1 x e 1 x r 1 x by a series of elementary row operations; Step 5 Substituting x by Θ 1r in polynomial rx we get a RFPrLR circulant matrix R rθ 1r AA {12} B If the last column of R isn t b then AX b has no solution Otherwise the RFPrLR circulant linear system AX b has a solution Substituting x by Θ 1r in polynomial cx ex we have two RFPrLR circulant matrices C cθ 1r A {12} B E eθ 1r A {12} A So the unique special solution of AX b is X 1 c c 1 c 0 + rc T the general solution of AX b is X 2 X 1 + I n EY Y is an arbitrary n-dimension column vector The advantage of the above algorithm is that it can solve AX b whether the coefficient matrix of AX b is singular or nonsingular By Lemma 3 Theorem 7 we have the following theorem Theorem 9 Let A RLPrFLcircfra a 1 a 0 be a nonsingular RLPrFL circulant matrix of order n over C b b b 1 b 0 T Then there exists a unique RFPrLR circulant matrix C RFPrLRcircfrc 0 c 1 c of order n over C such that the unique solution of AX b is X c 0 + rc c 1 c n 2 c T By Lemma 3 Theorem 8 we have the following theorem Theorem 10 Let A RLPrFLcircfra a 1 a 0 be a singular RLPrFL circulant matrix of order n over C b b b 1 b 0 T If the solution of AX b exists then there exists a unique RFPrLR circulant matrix C RFPrLRcircfrc 0 c 1 c a unique RFPrLR circulant matrix E RFPrLRcircfre 0 e 1 e of order n over C such that X 1 c 0 + rc c 1 c n 2 c T is the unique special solution of AX b X 2 X 1 + ÎnI n EY is the general solution of AX b Y is an arbitrary n-dimension column vector În is given in Equation 5 By Theorem 9 Theorem 10 we can get the fast algorithm for solving the RLPrFL circulant linear system AX b A RLPrFLcircfra a 1 a 0 X x 1 x 2 x n T b b b 1 b 0 T Step 1 From the RLPrFL circulant linear system AX b we get the polynomial fx a i x i gx x n rx 1 i0 bx b 0 rb + b i x i ; i1 Step 2 Change the polynomial matrix fx bx gx 0 dx cx by a series of elementary row 0 c 1 x operations; Step 3 If dx 1 then the RLPrFL circulant linear system AX b has a unique solution Substituting x by Θ 1r in polynomial cx we have a RFPrLR circulant matrix C cθ 1r RFPrLRcircfrc 0 c 1 c So the unique solution of AX b is c 0 + rc c 1 c n 2 c T ; Step 4 If dx 1 dividing gx by dx we get the quotient g 1 x change the polynomial matrix fxdx dxbx dxfx dxfxbx g 1 x E-ISSN: Issue 11 Volume 12 November 13

7 1 cx ex rx 0 c 1 x e 1 x r 1 x by a series of elementary row operations; Step 5 Substituting x by Θ 1r in polynomial rx we get a RFPrLR circulant matrix R rθ 1r AA {12} B If the last column of R isn t b then AX b has no solution Otherwise the RLPrFL circulant linear system AX b has a solution Substituting x by Θ 1r in polynomial cx ex we have two RFPrLR circulant matrices C cθ 1r A {12} B E eθ 1r A {12} A So the unique special solution of AX b is X 1 c 0 + rc c 1 c n 2 c T X 2 X 1 + ÎnI n EY is the general solution of AX b Y is an arbitrary n-dimension column vector 3 Examples Example 1 Solve the RFP3LR circulant linear system AX b A RFP3LRcircfr b T From A RFP3LRcircfr b T we get the polynomial Then fx 2 + x + x 3 gx 1 3x + x 4 bx 1 + 2x + x 2 Ax fx bx gx x + x x + x 2 1 3x + x 4 0 We transform the polynomial matrix Ax by a series of elementary row operations as follows: 2 + x + x x + x Ax 2 1 3x + x x1 2 + x + x x + x 2 1 5x x 2 p 3 x 1 + x2 2 5x x 2x 3 x 4 1 5x x 2 x 2x 2 x x p 1 x 1 5x x 2 x 2x 2 x x x p1 x x p 2x p 3 x x p 2x p 4 x x p 2x p 1 x 1 + 7x + 10x 2 + 3x 3 x 4 p 2 x x x x x x5 p 3 x x+105x 2 21x 3 43x 4 25x 5 p 4 x x+ 67 x x x x5 Since dx 1 then the RFP3LR circulant linear system AX b has a unique solution On the other h cx x x x x x5 Substituting x by Θ 13 in polynomial cx we have the RFP3LR circulant matrix C cθ 13 RFP3LRcircfr So the unique solution of AX b is the last column of C ie X T E-ISSN: Issue 11 Volume 12 November 13

8 Example 2 Solve the RFP6LR circulant linear system AX b A RFP6LRcircfr b T From A RFP6LRcircfr b T we get the polynomial fx x gx 1 6x + x 2 Then bx x Ax fx bx gx x 1 6x + x 2 0 We transform the polynomial matrix Ax by a series of elementary row operations as follows: x x Ax 1 6x + x x x x 0 q 1 x x q1 x 0 q 2 x q 1 x x q 2 x x 10 3x 2 It is obvious that dx x 1 it denoted that A is singular Then g 1 x gx/dx x dxfx x x 2 dxbx x + x 2 dxfxbx x + 3x x 3 Thus we structure matrix Bx transform the polynomial matrix Bx by a series of elementary row operations as follows: Bx fxdx dxbx dxfx dxfxbx g 1 x r 1 x s 1 x s 2 x s 3 x x x2 r 2 x s 1 x s 2 x s 3 x x s1 x s 2 x s 3 x x s 4 x s 5 x s 6 x x r 1 x x x 2 r 2 x x s 1 x x + x 2 s 2 x x x 2 s 3 x x+3x x 3 s 4 x s 5 x x x + 1 x2 s 6 x x x x3 Substituting x by Θ 16 in polynomial rx x x3 we get the RFP6LR circulant matrix R rθ 16 AA {12} B 10 3 x x RFP6LRcircfr E-ISSN: Issue 11 Volume 12 November 13

9 Since the last column of R is b the RFP6LR circulant linear system AX b has a solution Substituting x by Θ 16 in polynomial cx ex x we have the RFP6LR circulant matrices C cθ 16 A {12} B RFP6LRcircfr E eθ 16 A {12} A RFP6LRcircfr x + 1 x2 x So the unique special solution of AX b is the last column of C ie X T a general solution of AX b is X 2 X 1 + I n EY Y k 1 k 2 T C 10 k 2 k k k 2 Acknowledgements: The project is supported by the Development Project of Science & Technology of Shong Province Grant Nos 12GGX10115 the AMEP of Linyi University China References: [1] M Donatelli A multigrid for image deblurring with Tikhonov regularization Numer Linear Algebra Appl pp [2] Y Jing H Jafarkhani Distributed differential space-time coding for wireless relay networks IEEE Trans Commun pp [3] A Daher E H Baghious G Burel Fast algorithm for optimal design of block digital filters based on circulant matrices IEEE Signal Process Lett pp [4] T A Gulliver M Harada New nonbinary self-dual codes IEEE Trans Inform Theory pp [5] X Q Jin V K Sin L L Song Circulantblock preconditioners for solving ordinary differential equations Appl Math Comput pp [6] P Davis Circulant Matrices Wiley New York 1979 [7] Z L Jiang Z X Zhou Circulant Matrices Chengdu Technology University Publishing Company Chengdu 1999 in Chinese [8] J Gutirrez-Gutirrez Positive integer powers of complex symmetric circulant matrices Appl Math Comput pp [9] J Rimas On computing of arbitrary positive integer powers for one type of even order symmetric circulant matrices-ii Appl Math Comput pp [10] J Rimas On computing of arbitrary positive integer powers for one type of even order symmetric circulant matrices-i Appl Math Comput pp [11] J Rimas On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices-ii Appl Math Comput pp [12] J Rimas On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices-i Appl Math Comput pp [13] C David Regular representations of semisimple algebras separable field extensions group characters generalized circulants generalized cyclic codes Linear Algebra Appl pp [14] Z L Jiang Z B Xu Efficient algorithm for finding the inverse group inverse of FLS r- circulant matrix J Appl Math Comput pp [15] Z L Jiang D H Sun Fast Algorithms for Solving the Inverse Problem of Ax b Proceedings of the Eighth International Conference on Matrix Theory Its Applications in China 08 pp [16] Z L Jiang Fast Algorithms for Solving FLS r- Circulant Linear Systems SCET 12 Xi an pp [17] Z L Jiang Z W Jiang On the Norms of RF- PLR Circulant Matrices with the Fibonacci Lucas Numbers SCET 12Xi an pp E-ISSN: Issue 11 Volume 12 November 13

10 [18] J Li Z L Jiang N Shen Explicit determinants of the Fibonacci RFPLR circulant Lucas RFPLL circulant matrix JP J Algebra Number Theory Appl pp [19] Z L Jiang J Li N Shen On the explicit determinants of the RFPLR RFPLL circulant matrices involving Pell numbers in information theory ICICA 12 Part II Commun Comput Inf Sci pp [] Z P Tian Fast algorithm for solving the first plus last circulant linear system J Shong Univ Nat Sci pp [21] N Shen Z L Jiang J Li On explicit determinants of the RFMLR RLMFL circulant matrices involving certain famous numbers WSEAS Trans Math pp [22] Z L Jiang N Shen J Li On the explicit determinants of the RFMLR RLMFL circulant matrices involving Jacobsthal numbers in communication ICICA 13 Commun Comput Inf Sci [23] ZPTian Fast algorithms for solving the inverse problem of AX b in four different families of patterned matrices Far East J Appl Math pp 1 12 [24] Z L Jiang Z B Xu S P Gao Algorithms for finding the inverses of factor block circulant matrices Numer Math J Chinese Univ English Ser pp 1 11 [25] Z L Jiang Z B Xu A new algorithm for computing the inverse generalized inverse of the scaled factor circulant matrix J Comput Math pp [26] Z L Jiang S Y Liu Efficient algorithms for computing the minimal polynomial the inverse of level-kπ-circulant matrices Bull Korean Math Soc 3 03 pp [27] Z L Jiang S Y Liu Level-m scaled circulant factor matrix over the complex number field the quaternion division algebra J Appl Math Comput pp [28] S G Zhang Z L Jiang S Y Liu An application of the Gröbner basis in computation for the minimal polynomials inverses of block circulant matrices Linear Algebra Appl pp [29] J Gutirrez-Gutirrez Positive integer powers of complex skew-symmetric circulant matrices Appl Math Comput pp [30] N L Tsitsas E G Alivizatos G H Kalogeropoulos A recursive algorithm for the inversion of matrices with circulant blocks Appl Math Comput pp [31] S M El-Sayed A direct method for solving circulant tridiagonal block systems of linear equations Appl Math Comput pp [32] W F Trench Properties of multilevel block α- circulants Linear Algebra Appl pp [33] W F Trench Properties of unilevel block circulants Linear Algebra Appl pp [34] M Abreu D Labbate R Salvi N Z Salvi Highly symmetric generalized circulant permutation matrices Linear Algebra Appl pp [35] N V Davydova O Diekmann S A V Gils On circulant populations I The algebra of semelparity Linear Algebra Appl pp [36] M Andrecut Applications of left circulant matrices in signal image processing Modern Phys Lett B pp [37] R E Cline R J Plemmons G Worm Generalized inverses of certain Toeplitz matrices Linear Algebra Appl pp [38] G R Wang Y M Wei S Z Qiao Generalized Inverses: Theory Computations Science Press Beijing/New York 04 [39] T N E Greville Some new generalized inverses with spectral properties Proc Of the Conf On Generalized matrix inverses Texas Tech Univ March 1968 [] J E Scroggs P L Odell An alternate definition of a pseudo-inverse of a matrix J Soc Ind Appl Math pp E-ISSN: Issue 11 Volume 12 November 13

Exact Determinants of the RFPrLrR Circulant Involving Jacobsthal, Jacobsthal-Lucas, Perrin and Padovan Numbers

Tingting Xu Zhaolin Jiang Exact Determinants of the RFPrLrR Circulant Involving Jacobsthal Jacobsthal-Lucas Perrin and Padovan Numbers TINGTING XU Department of Mathematics Linyi University Shuangling