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 Road Linyi city CHINA xutingting655@6com ZHAOLIN JIANG Department of Mathematics Linyi University Shuangling Road Linyi city CHINA jzh08@sinacom Abstract: The row first-plus-rlast r-right RFPrLrR circulant matrix and row last-plus-rfirst r-left RLPrFrL circulant matrices are two special pattern matrices On the basis of the beautiful properties of famous numbers and the inverse factorization of polynomial we give the exact determinants of the two pattern matrices involving Jacobsthal Jacobsthal-Lucas Perrin and Padovan numbers Key Words: Determinant; RFPrLrR circulant matrix; RLPrFrL circulant matrix; Jacobsthal number; Jacobsthal- Lucas number; Perrin number; Padovan number; Introduction Circulant matrix family occurs in various fields applied in image processing communications signal processing encoding and preconditioner Meanwhile the circulant matrices have been extended in many directions recently The fx-circulant matrix is another natural extension of the research category please refer to Recently some authors researched the circulant type matrices with famous numbers In Shen et al discussed the explicit determinants of the RFMLR and RLMFL circulant matrices involving certain famous numbers Jaiswal 4 showed some determinants of circulant matrices whose elements are the generalized Fibonacci numbers Lind presented the determinants of circulant and skew circulant involving Fibonacci numbers in 5 Gao et al 6 considered the determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas numbers Akbulak and Bozkurt 7 proposed some properties of Toeplitz matrices involving Fibonacci and Lucas numbers In 8 authors considered circulant matrices with Fibonacci and Lucas numbers and proposed their explicit determinants and inverses by constructing the transformation matrices Jiang and Hong gave exact determinants of some special circulant matrices involving four kinds of famous numbers in 9 See more of the literatures in 0 We introduce two new patten matrices ie row com Corresponding author E-mail addresses: jzh08@sina first-plus-r last r-right RFPrLrR circulant matrix and row last-plus-r first r-left RLPrFrL circulant matrix Based on the characteristic polynomial of the basic RFPrLrR circulant matrix and the Binet formulae of famous numbers we obtain the exact determinants of two patten matrices with Jacobsthal Jacobsthal-Lucas Perrin and Padoven numbers A row first-plus-rlast r-right RFPrLrR circulant matrix with the first row a 0 a a n denoted by RFPrLRcirc r fr a 0 a a n is meant to be a square matrix of the form A a 0 a a n ra n a 0 ra n a n ra n ra n ra n a n ra ra ra a 0 ra n Be aware that the RFPrLrR circulant matrix is a x n rx r-circulant matrix We define Θ rr as the basic RFPrLrR circulant matrix that is Θ rr 0 0 0 0 0 0 0 0 0 0 0 0 r r 0 0 0 RFPrLRcirc r fr0 0 0 n n Both the minimal polynomial and the characteristic polynomial of Θ rr are gx x n rxr which has only simple roots denoted by π i i n E-ISSN: 4-880 5 Volume 5 06
Tingting Xu Zhaolin Jiang A row last-plus-rfirst r-left RLPrFrL circulant matrix with the first row a 0 a a n denoted by RLPrFLcirc r fr a 0 a a n is meant to be a square matrix of the form B a 0 a n a n a a n ra 0 ra 0 a ra 0 ra ra a n ra 0 ra n ra n ra n Let ARLPrFLcirc r fra 0 a a n and B RFPrLRcirc r fra n a n a 0 By explicit computation we find A BÎn În is the backward identity matrix of the form Î n 0 0 The Jacobsthal sequences {J n } Jacobsthal- Lucas sequences {j n } 4 5 Perrin sequences {R n } and Padovan sequences {P n } 6 7 8 are defined by the following recurrence relations respectively: J n J n J n n j n j n j n n 4 R n R n R n n 5 P n P n P n n 6 with the initial condition J 0 0 J j 0 j R 0 R 0 R and P 0 P P The first few members of these sequences are given as follows: n 0 4 5 6 J n 0 5 j n 5 7 7 65 R n 0 5 5 P n 4 The sequences {J n } and {j n } are given by the Binet formulae J n αn β n α β and j n α n β n α β are the roots of the equation x x 0 While recurrences 5 and 6 involve the characteristic equation x x 0 its roots are denoted by r r r Then we obtain: r r r 0 r r r r r r r r r 7 In addition the Binet form for the Perrin sequences is R n r n r n r n 8 and the Binet form for Padovan sequences is P n a r n a r n a r n 9 a i j j i Main Results r j r i r j i By Proposition 5 in 9 and properties of RFPrLrR circulant matrices we deduce the following lemma Lemma Let A RFPrLRcirc r fra 0 a a n Then the eigenvalues of A are given by n λ i fπ i a j π j i i 0 n j0 π i i n are the eigenvalues of Θ rr and the determinant of A is given by det A λ i n a j π j i j0 Lemma Assume π i i n are the roots of the characteristic polynomial of Θ rr If a 0 then aπi bπi cπ i d bπi cπ i d d n b n dr s n t n br s n t n r d c b E-ISSN: 4-880 6 Volume 5 06
Tingting Xu Zhaolin Jiang a b c R and s c c 4bd b If a 0 then aπi bπi cπ i d ; t c c 4bd b d n an d rx n rxn r X n an rxn r X n rxn r X n a n r a b c d br X n X n x n xn xn and x x x are the roots of the equation aπ i bπ i cπ i d 0 Proof Since π i i n are the roots of the characteristic polynomial of Θ rr gx x n rx r can be factored as x n rx r x π i Let x x x be the roots of the equation aπ i bπ i cπ i d 0 If a 0 please see for details of the proof If a 0 then aπi bπi cπ i d n a n πi b a π i c a π i d a n a n π i x π i x π i x a n n x π i x π i x π i a n x n rx rx n rx rx n rx r a n{ x x x n rx x x x x n x x n x x n r x x n x x n x x n r x x x x n x n x n r x n x x x n x x x n x x r x n x n x n r x x x r x x x x x x r x x x r } Let X n x n xn xn we obtain x x n x x n x x n X nx n from x n xn x n x n xn xn x x n x x n x x n Taking the relation of roots and coefficients x x x b a x x x x x x c a x x x d a into account we get that aπi bπi cπ i d d n an d rx n rxn r X n an rxn r X n rxn r X n a n r a b c d br X n X n x n xn xn and x x x are the roots of the equation aπi bπ i cπ i d 0 In the following we show the exact determinants of the RFPrLrR and RLPrFrL circulant matrices involving some related famous numbers Determinants of the RFPrLrR and RLPrFrL Circulant Matrices Involving the Jacobsthal Numbers Theorem If A RFPrLRcirc r fr J 0 J n then deta n n rjn rjn r J n s n t n rj n rj n n rjn r J n s n t n r rj n rj n s rj n rj n J n rjn rj n 8r J n J n J n t rj n rj n J n rjn rj n 8r J n J n J n E-ISSN: 4-880 7 Volume 5 06
Tingting Xu Zhaolin Jiang Proof The matrix A can be written as J 0 J J n rj n J 0 rj n J n A rj rj rj J rj rj rj J 0 rj n Using the Lemma the determinant of A is deta J0 J π i J n πi n α β α β π i α β α β π i αn β n πi n α β rjn π i rj n rj n πi rj n π i πi According to Lemma we can get rjn π i rj n rj n πi rj n n n rj n rjn r J n s n t n r J n s n t n r Then we obtain that deta n n rjn rjn r J n s n t n rj n rj n n rjn r J n s n t n r rj n rj n s rj n rj n J n rjn rj n 8r J n J n J n t rj n rj n J n rjn rj n 8r J n J n J n Using the method in Theorem similarly we also have Theorem 4 If A RFPrLRcirc r fr J n J 0 then deta r Jn n r Jn r n r Jn n rj n rj n r J n r n n rj n rj n Theorem 5 If J RLPrFLcirc r fr J 0 J n then we have n r Jn r Jn r n r Jn detj n rj n rj n r J n r n n nn rj n rj n Proof The matrix J can be written as J 0 J n J n J J n rj 0 rj 0 J J n rj n4 rj n rj n J n rj 0 rj n rj n rj n J n J n J 0 rj 0 J n rj 0 J rj n rj n4 rj n J n rj n rj n rj n J n rj 0 0 0 0 0 0 0 0 0 0 0 0 0 then we can get detj deta detγ which matrix A RFPrLRcirc r fr J n J 0 its determinant could be obtained through Theorem deta and r Jn n r Jn r n r Jn n rj n rj n r J n r n n rj n rj n detγ nn E-ISSN: 4-880 8 Volume 5 06
Tingting Xu Zhaolin Jiang So detj deta detγ n r Jn r Jn r n r Jn n rj n rj n r J n r n n nn rj n rj n r s j n r jn 8r 6rj n r t j n r jn 8r 6rj n Theorem 8 If J RLPrFLcirc r fr j 0 j n then we have 4 Determinants of the RFPrLrR and RLPrFrL Circulant Matrix Involving Jacobsthal-Lucas Numbers Theorem 6 If B RFPrLRcirc r fr j 0 j n then n rjn detb rj n rj n n rjn r rjn s n t n rj n rj n n rjn r s n tn r rj n rj n s rj n rj n j n rjn rj n 6rjn 8r j n j n j n t rj n rj n j n rjn rj n 6rjn 8r j n j n j n Similarly we also get the following Theorem 7 Theorem 7 If B RFPrLRcirc r fr j n j 0 then n r detb jn n rj n rj n nr r rj n s n t n n rj n rj n nr r s n tn r j n j n n rj n rj n n r detj jn n rj n rj n nr r rj n s n t n n rj n rj n n r r s n tn r j n j n n rj n rj n nn r s j n r jn 8r 6rj n r t j n r jn 8r 6rj n Proof The matrix J can be written as j 0 j n j n j j n rj 0 rj 0 J j n rj n4 rj n rj n j n rj 0 rj n rj n rj n j n j n j 0 rj 0 j n rj 0 j rj n rj n4 rj n j n rj n rj n rj n j n rj 0 0 0 0 0 0 0 0 0 0 0 0 0 then we can get detj detb detγ E-ISSN: 4-880 9 Volume 5 06
Tingting Xu Zhaolin Jiang with matrix B RFPrLRcirc r fr j n j 0 its determinant could be obtained through Theorem 7 n r detb jn n rj n rj n nr r rj n s n t n n rj n rj n nr r s n tn r j n j n n rj n rj n r s j n r jn 8r 6rj n r t j n r jn 8r 6rj n and detγ nn So detj detb detγ n r jn n rj n rj n nr r rj n s n t n n rj n rj n n r r s n tn r j n j n n rj n rj n nn r s j n r jn 8r 6rj n r t j n r jn 8r 6rj n 5 Determinants of the RFPrLrR and RLPrFrL Circulant Matrix Involving Perrin Numbers Theorem 9 Let C RFPrLRcirc r fr R R R n Then det C rr n rr n n n rrn rrn rr n rxn rx n r X n n rrn rx n rx n r X n r X n Y r rr n rrn rr n r X n rr n n ry n ry n ry n ry n r Y n r Y n r X n x n xn xn and x x x are the roots of the equation rr n x rr n rr n x rr n x rrn rr n 0 Y n y n yn yn y y y are the roots of the equation y y 0 Proof Obviously the matrix C has the form R R R n rr n R rr n R n C rr rr 4 rr R rr rr rr R rr n Owing to Lemma the Binet form 8 and 7 det C R R π i R n πi n n k j j r k j π k i r j rj nπn i r j π i rrn πi rr n rr n πi πi π i rrn πi rr n rr n π i π i E-ISSN: 4-880 0 Volume 5 06
Tingting Xu Zhaolin Jiang By Lemma and the recurrence 5 we obtain rr n π i rr n rr n π i rr n πi rr n rr n n rr n rr n n rrn rrn rr n rxn rx n r X n n rrn rx n rx n r X n r X n n rr n r rr n rr n rr n r X n X n x n xn xn and x x x are the roots of the equation rr n x rr n rr n x rrn x rrn rr n 0 And πi πi ry n ryn ry n ry n r Y n r Y n r Y n y n yn yn and y y y are the roots of the equation y y 0 Consequently det C rr n rr n n n rrn rrn rr n rxn rx n r X n n rrn rx n rx n r X n r X n Y r rr n rrn rr n r X n rr n n Theorem 0 Let C RFPrLRcirc r fr R n R n R Then det C R n r n rrn rx n x n rr n rx n xn r R n R n R n R n r n r ry n ry n r Y n ry n ry n r Y n r Y n x 5r R n χ R n x 5r R n χ R n χ R n 5r 4R n rr n r Y n y n yn yn and y y y are the roots of the equation y y 0 Proof The matrix C has the form R n R n R rr R n rr R rr n rr n rr n R n rr n rr n rr n R n rr According to Lemma the Binet form 8 and 7 we have det C Rn R n π i R πi n n k0 j r nk j π k i j r n j r j π n i r j π i R n rπi R n 5rπ i R n r πi π i Using Lemma and the recurrence 5 we obtain Rn rπi R n 5rπ i R n r R n r n rr n rx n x n rr n rx n x n r R n R n R n R n r n E-ISSN: 4-880 Volume 5 06
Tingting Xu Zhaolin Jiang x 5r R n χ R n x 5r R n χ R n χ R n 5r 4R n rr n r And πi π i r ry n ry n r Y n ry n ry n r Y n r Y n Y n y n yn yn and y y y are the roots of the equation y y 0 Therefore det C R n r n rrn rx n x n rr n rx n xn r R n R n R n R n r n Theorem Let R RLPrFLcirc r fr R R R n Then { Rn r n rrn rx n x n det R rr n rx n xn r } R n R n R n R n r n nn r ry n ry n r Y n ry n ry n r Y n r Y n x 5r R n χ R n x 5r R n χ R n χ R n 5r 4R n rr n r Y n y n yn yn and y y y are the roots of the equation y y 0 Proof Since det În nn the result can be derived from Theorem 0 and the relation 6 Determinants of the RFPrLrR and RLPrFrL Circulant Matrix Involving Padovan Numbers Theorem Let D RFPrLRcirc r frp P P n Then detd n P rp n rpn n P rp n ru n ru n r U n n rpn ru n ru n r U n r U n n rpn r P P a a a a a a rp n rp n r U n r n n r V n n n r V n V n Un u n un u n u u u are the roots of the equation rp n x a a a rp n rp n x P rp n rp n xp rp n 0 and V n v nvn vn v v v are the roots of the equation y y 0 Proof The matrix D has the form P P P n rp n P rp n P n D rp rp 4 rp P rp rp rp P rp n det D n k j j P P π i P n πi n a j r k j π k i a j r j rj nπn i r j π i rpn πi a a a rp n rp n π i πi π i P rp n rp n πi P rp n πi π i from Lemma the Binet form 9 and 7 Using E-ISSN: 4-880 Volume 5 06
Tingting Xu Zhaolin Jiang Lemma and the recurrence 6 we obtain rp n π i a a a rp n rp n π i P rp n rp n πi P rp n n n P rp n rpn P rp n run ru n r U n n rpn ru n ru n r U n r U n n rp n r P P a a a and a a a rp n rp n r U n π i πi r n n r V n n n r V n V n U n u n un un u u u are the roots of the equation rp n x a a a rp n rp n x P rp n rp n xp rp n 0 We have the following results: detd n P rp n rpn n P rp n ru n ru n r U n n rpn ru n ru n r U n r U n n rpn r P P a a a a a a rp n rp n r U n r n n r V n n n r V n V n Un u n un u n u u u are the roots of the equation rp n x a a a rp n rp n x P rp n rp n xp rp n 0 and V n v n vn vn v v v are the roots of the equation y y 0 Theorem Let D RFPrLRcirc r frp n P n P Then detd Pn r n a a a rn P n r a a a ru n ru n r U n rn ru n ru n r U n r U n rn r P n P n P n rp P n rp rp r U n r n n r V n n n r V n r V n U n u n un un u u u are the roots of the equation rp x P n rp rp x P n r a a a rp xpn r a a a 0 and V n v n vn vn v v v are the roots of the equation y y 0 Proof The matrix D has the form P n P n P rp P n rp P rp n rp n rp n P n rp n rp n rp n P n rp n According to Lemma the Binet form 9 and 7 we have det D n k0 j Pn P n π i P πi n a j r nk j π k i j a j r n j rπ i P n r πi πi π i Pn r a a a r πi πi π i P n r a a a πi π i Using Lemma and 9 we obtain a j r j π n i r j π i E-ISSN: 4-880 Volume 5 06
Tingting Xu Zhaolin Jiang rπi P n r πi P n r a a a r πi P n r a a a P n r n n a a a rp Pn r a a a ru n ru n r U n n rp ru n ru n r U n r U n n rp r P n P n P n rp P n rp rp r U n U n u n un un u u u are the roots of the equation rp x P n rp rp x Pn r a a a rp x Pn r a a a 0 π i π i r n n r V n n n r V n r V n V n v n vn vn v v v are the roots of the equation y y 0 Then we have the following results: det D Pn r n a a a rn P n r a a a ru n ru n r U n rn ru n ru n r U n r U n rn r P n P n P n rp P n rp rp r U n r n n r V n n n r V n r V n U n u n un un u u u are the roots of the equation rp x P n rp rp x P n r a a a rp xpn r a a a 0 and V n v n vn vn v v v are the roots of the equation y y 0 Theorem 4 Let P RLPrFLcirc r fr P P P n Then det P { Pn r n a a a rn P n r a a a ru n ru n r U n rn ru n ru n r U n r U n rn r P n P n P n rp P n rp rp r U n } nn r n n r V n n n r V n r V n U n u n un un u u u are the roots of the equation rp x P n rp rp x P n r a a a rp xpn r a a a 0 and V n v n vn vn v v v are the roots of the equation y y 0 Proof The theorem can be proved by using Theorem and the relation 7 Conclusion In this paper we introduce RFPrLrR circulant matrix and RLPrFrL circulant matrices which are two special pattern matrices Based on the Binet formulaes of famous numbers and the characteristic polynomial of the basic RFPrLrR circulant matrix we give the exact determinants of the two pattern matrices involving Jacobsthal Jacobsthal-Lucas Perrin and Padovan numbers in section 4 5 and 6 Acknowledgements: The research is supported by the Development Project of Science & Technology of Shandong Province Grant No 0GGX05 and the AMEP of Linyi University China References: P J Davis Circulant matrices Wiley & Sons New York 979 Z L Jiang and Z X Zhou Circulant matrices Chengdu Technology University Publishing Company Chengdu 999 N Shen Z L Jiang and J Li On explicit determinants of the RFMLR and RLMFL circulant matrices involving certain famous numbers E-ISSN: 4-880 4 Volume 5 06
Tingting Xu Zhaolin Jiang WSEAS Transactions on Mathematics 0 4 5 4 D Jaiswal On determinants involving generalized Fibonacci numbers Fibonacci Quart 7 969 pp 9 0 5 D A Lind A Fibonacci circulant Fibonacci Quart 8 5 970 pp 449 455 6 Y Gao Z L Jiang and Y P Gong On the determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas numbers WSEAS Transactions on Mathematics 4 0 47 48 7 M Akbulak and D Bozkurt On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers Hacet J Math Stat 7 008 pp 89 95 8 S Q Shen J M Cen and Y Hao On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers Appl Math Comput 7 0 pp 9790 9797 9 X Y Jiang and K Hong Exact determinants of some special circulant matrices involving four kinds of famous numbers Abstract and Applied Analysis 04 04 Article ID 7680 pages 0 D Lin Fibonacci-Lucas quasi-cyclic matrices Fibonacci Quart 40 00 pp 80 86 T T Xu Z L Jiang and Z W Jiang Explicit Determinants of the RFPrLrR circulant and RLPrFrL circulant matrices involving some famous numbers Abstract and Applied Analysis 04 Article ID 64700 9 pages Z L Jiang J Li and N Shen On the explicit determinants and singularities of r-circulant and left r-circulant matrices with some famous numbers WSEAS Transactions on Mathematics 0 4 5 A F Horadam Jacobsthal representation numbers Fibinacci Quart 4 996 pp 40 54 4 Z Čerin Sums of squares and products of Jacobsthal numbers J Integer Seq 0 007 pp 5 5 P Barry Triangle geometry and Jacobstahal numbers Irish Math Sco Bulletin 5 00 pp 45 57 6 F Yilmaz and D Bozkurt Hessenberg matrices and the Pell and Perrin numbers Journal of Number Theory 8 0 pp 90 96 7 J Blmlein D J Broadhurst and J A M Vermaseren The multiple zeta value data minem Computer Physics Communications 8 00 pp 58 65 8 M Janjić Determinants and recurrence sequences Journal of Integer Sequences 5 0 pp 9 D Chillag Regular representations of semisimple algebras separable leld extensions group characters generalized circulants and generalized cyclic codes Linear Algebra Appl 8 995 pp 47 8 E-ISSN: 4-880 5 Volume 5 06