Computing the Determinant and Inverse of the Complex Fibonacci Hermitian Toeplitz Matrix

Transcription

1 British Journal of Mathematics & Computer Science 9(6: ; Article nobjmcs30398 ISSN: SCIENCEDOMAIN international wwwsciencedomainorg Computing the Determinant and Inverse of the Complex Fibonacci Hermitian Toeplitz Matrix Jixiu Sun 2 and Zhaolin Jiang 2 School of Mathematics and Statistics Shandong Normal University Jinan PR China 2 School of Sciences Linyi University Linyi PR China Authors contributions This work was carried out in collaboration by both authors Author ZJ designed the study proposed the concerned problem Author JS managed the analyses of the study and wrote the first draft of the manuscript Both authors read and approved the final manuscript Article Information DOI: 09734/BJMCS/206/30398 Editor(s: ( Morteza Seddighin Indiana University East Richmond USA Reviewers: ( Elif etin Manisa Celal Bayar University Turkey (2 Cherng-tiao Perng Norfolk State University USA Complete Peer review History: Received: 7 th November 206 Accepted: 26 th November 206 Original Research Article Published: 30 th November 206 Abstract In this paper we consider the determinant and the inverse of the complex Fibonacci Hermitian Toeplitz matrix We first give the definition of the complex Fibonacci Hermitian Toeplitz matrix Then we compute the determinant and inverse of the complex Fibonacci Hermitian Toeplitz matrix by constructing the transformation matrices Keywords: Hermitian Toeplitz matrix; determinant; inverse; complex Fibonacci numbers 200 Mathematics Subject Classification: 5A09 5A5 5B05 B39 *Corresponding author: jzh208@sinacom; Co-author: sunjixiu34@63com;

2 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 Introduction Studies show that there has been an increasing interest on Fibonacci sequence and its generalizations One of them is the concept of complex Fibonacci numbers Fn which was defined by Horadam [] The n-th complex Fibonacci number is given by the equality Fn = F n + if n+ where i is the imaginary unit which satisfies i 2 = Complex Fibonacci numbers satisfy the same recurrence relation Fn = Fn +F n 2(n 2 of the classical Fibonacci numbers with different initial conditions ie = i F = + i On the other hand Hermitian Toeplitz matrices have important applications in various disciplines including image processing signal processing and solving least squares problems [2 3 4] It is an ideal research area and hot topic for the inverses of the special matrices with famous numbers Some scholars showed the explicit determinants and inverses of the special matrices involving famous numbers Lin showed the determinant of the Fibonacci-Lucas quasi-cyclic matrices in [5] Circulant matrices with Fibonacci and Lucas numbers are discussed and their explicit determinants and inverses are proposed in [6] The authors provided determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers in [7] In [8] circulant type matrices with the k- Fibonacci and k-lucas numbers are considered and the explicit determinants and inverse matrices are presented by constructing the transformation matrices The explicit determinants of circulant and left circulant matrices including Tribonacci numbers and generalized Lucas numbers are shown based on Tribonacci numbers and generalized Lucas numbers in [9] In [0] Jiang and Hong gave the exact determinants of the RSFPLR circulant matrices and the RSLPFL circulant matrices involving Padovan Perrin Tribonacci and the generalized Lucas numbers by the inverse factorization of polynomial Jiang et al [] gave the invertibility of circulant type matrices with the sum and product of Fibonacci and Lucas numbers and provided the determinants and the inverses of these matrices It should be noted that Jiang and Zhou [2] obtained the explicit formula for spectral norm of an r-circulant matrix whose entries in the first row are alternately positive and negative and the authors [3] investigated explicit formulas of spectral norms for g-circulant matrices with Fibonacci and Lucas numbers Furthermore in [4] the determinants and inverses are discussed and evaluated for Tribonacci skew circulant type matrices The authors [5] proposed the invertibility criterium of the generalized Lucas skew circulant type matrices and provided their determinants and the inverse matrices The determinants and inverses of Tribonacci circulant type matrices are discussed in [6] Determinants and inverses of Fibonacci and Lucas skew symmetric Toeplitz matrices are given by constructing the special transformation matrices in [7] The purpose of this paper is to obtain better results for the determinants and inverses of complex Fibonacci Hermitian Toeplitz matrix In this paper we adopt the following two conventions 0 0 = i 2 = and we define a kind of special matrix as follows Definition A complex Fibonacci Hermitian Toeplitz matrix is a square matrix of the form 0 F Fn 4 Fn 3 F n 2 0 Fn 5 Fn 4 F n 3 F 0 Fn 6 Fn 5 F n 4 T F n = ( F n 4 F n 5 F n 6 0 F F n 3 F n 4 F n 5 0 F n 4 F 0 F n 2 F n 3 where F 0 F F n 2 are the complex Fibonacci numbers It is evidently determined by its first row and first column and T F n = T T F n n n 2

3 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 Lemma Let p i be a Hessenberg matrix κ 2 κ 0 0 κ 3 κ 2 κ κ 4 κ 3 κ 2 κ p i = κ 5 κ 4 κ 3 κ 2 0 κ κ i+ κ 5 κ 4 κ 3 κ 2 then the determinant of p i can be expressed as i i det p i = i ( i+j κ i j+2κ i j det p j i 2 where the initial values of det p i are det p 0 = and det p = κ 2 Proof By using the Laplace expansion of matrix p i along the last row it is easy to check that det p i = ( i+ κ i+κ i det p 0 + ( i+2 κ iκ i 2 det p + ( i+3 κ i κ i 3 det p ( i+i κ 3κ det p i 2 + ( i+i κ 2κ 0 det p i i = ( i+j κ i j+2κ i j det p j for i 2 The initial values of det p i are det p 0 = and det p = κ 2 Lemma 2 Define an i i Toeplitz-like matrix by µ i µ i µ i 2 µ i 3 µ i 4 µ 2 µ α 2 α 0 0 α 3 α 2 α i([µ k ] i α 4 α 3 α 2 α α α 2 α i = α 5 α 4 α 3 α 2 α α4 α 3 α 2 α 0 α i α 5 α 4 α 3 α 2 α we have det i([µ k ] i α α 2 α i = i ( +j µ i+ jα i j det p j i i where det p j(0 j i is the determinant of Hessenberg matrix and can be calculated by Lemma 3

4 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 Proof By using the Laplace expansion of matrix i([µ k ] i α α 2 α i along the first row it is easy to check that det i([µ k ] i α α 2 α i = ( + µ iα i det p 0 + ( +2 µ i α i 2 det p + ( +3 µ i 2α i 3 det p ( +i µ 2α det p i 2 + ( +i µ α 0 det p i i = ( +j µ i j+α i j det p j where det p j(0 j i is the determinant of Hessenberg matrix and can be calculated by Lemma 2 Determinant and Inverse of the Complex Fibonacci Hermitian Toeplitz Matrix Let T F n be a complex Fibonacci Hermitian Toeplitz matrix In this section we first give the determinant of the matrix T F n and then we find the inverse of T F n assuming that it is invertible Theorem 2 Let T F n be a complex Fibonacci Hermitian Toeplitz matrix as the form of ( Then we have and det T F = 0 det T F 2 = det T F 3 = 2 det T F 4 = 6 det T F 5 = 2 det T F n = F 0 [F 0 (K 3 det n 3([η k ] n 3 α t 0 t t n 6 K 4 det n 3([ξ k ] n 3 α t 0 t t n 6 F n 3(K det n 3([η k ] n 3 α t 0 t t n 6 K 4 det n 3([F k ] n 3 α t 0 t t n 6 + F n 4(K det n 3([ξ k ] n 3 α t 0 t t n 6 K 3 det n 3([F k ] n 3 α t 0 t t n 6] n > 5 (2 where n 2 n 2 n 2 K = Fk n 2 k K 3 = ξ k n 2 k K 4 = η k n 2 k 0 = is one root of the characteristic equation x 2 + αx + t 0 = 0 i 2 2 = 2 i = α i + t k i 2 k (3 i n 3 ξ i = xf i + F n i 3 ( i n 3 ξ n 2 = xf n 3 η i = yf i + F n 4 i ( i n 4 η n 3 = yf n 4 η n 2 = yf n 3 + F 0 α = 2i t 0 = F2 t i = 2Fi ( i n 5 x = F n 2 y = F n 3 det n 3([η k ] n 3 α t 0 t t n 6 det n 3([ξ k ] n 3 α t 0 t t n 6 and det n 3([F k ] n 3 α t 0 t t n 6 can be calculated by Lemma 2 4

5 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 Proof Let T F n be a complex Fibonacci Hermitian Toeplitz matrix We can easily get the following conclusions: det T F = 0 det T F 2 = det T F 3 = 2 det T F 4 = 6 det T F 5 = 2 We can introduce the following two transformation matrices when n > 5 M = x y n n and N = n n 4 0 n n n where x = F n 2 F 0 y = F n 3 0 = is one root of the characteristic equation x 2 + αx + t 0 = 0 i 2 2 = 2 i = α i + t k i 2 k (3 i n 3 α = 2i t 0 = F 2 t i = 2F i ( i n 5 By using M N and the recurrence relations of the complex Fibonacci sequences the matrix T F n 5

6 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 is changed into the following form 0 K Fn 3 Fn 4 Fn 5 Fn 6 F2 F 0 K 2 Fn 4 Fn 5 Fn 6 Fn 7 F 0 F n 3 K 3 ξ n 3 ξ n 4 ξ n 5 ξ n 6 ξ 2 ξ F n 4 K 4 η n 3 η n 4 η n 5 η n 6 η 2 η 0 0 α 0 0 M T F n N = t 0 α t t 0 α t n 6 t t 0 α where n n ξ i = xfi + F n i 3 ( i n 3 ξ n 2 = xfn 3 η i = yfi + F n 4 i ( i n 4 η n 3 = yfn 4 η n 2 = yfn 3 + α = 2i t 0 = F2 t i = 2Fi ( i n 5 n 2 n 3 n 2 n 2 K = Fk n 2 k K 2 = Fk n 3 k K 3 = ξ k n 2 k K 4 = η k n 2 k By using the Laplace expansion of matrix M T F n N along the first column we can get that where det(m T F n N = F 0 [F 0 (K 3 det n 3([η k ] n 3 α t 0 t t n 6 K 4 det n 3([ξ k ] n 3 α t 0 t t n 6 F n 3(K det n 3([η k ] n 3 α t 0 t t n 6 K 4 det n 3([F k ] n 3 α t 0 t t n 6 + F n 4(K det n 3([ξ k ] n 3 α t 0 t t n 6 K 3 det n 3([F k ] n 3 α t 0 t t n 6] n 3([η k ] n 3 α t 0 t t n 6 η n 3 η n 4 η n 5 η n 6 η 4 η 3 η 2 η α 0 0 t 0 α t t 0 α = t 2 t t 0 α α t0 α 0 t n 6 t n 7 t t 0 α (n 3 (n 3 6

7 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 n 3([ξ k ] n 3 α t 0 t t n 6 ξ n 3 ξ n 4 ξ n 5 ξ n 6 ξ 4 ξ 3 ξ 2 ξ α 0 0 t 0 α t t 0 α = t 2 t t 0 α α t0 α 0 t n 6 t n 7 t t 0 α (n 3 (n 3 n 3([Fk ] n 3 α t 0 t t n 6 Fn 3 Fn 4 Fn 5 Fn 6 F4 F3 F2 F α 0 0 t 0 α t t 0 α = t 2 t t 0 α α t0 α 0 t n 6 t n 7 t t 0 α (n 3 (n 3 and the determinant of them can be calculated by Lemma 2 While we can obtain det T F n det M = det N = ( (n+(n 2 2 as (2 which completes the proof Theorem 22 Let T F n be an invertible complex Fibonacci Hermitian Toeplitz matrix and n > 6 Then we have T F n = ψ ψ 2 ψ 3 ψ n 2 ψ n ψ n ψ 2 ψ 22 ψ 23 ψ 2n 2 ψ 2n ψ n ψ 3 ψ23 ψ 33 ψ 3n 3 ψ 2n 2 ψ n 2 ψ n 2 ψ2n 2 ψ3n 3 ψ 33 ψ 23 ψ 3 ψ n ψ2n ψ2n 2 ψ23 ψ 22 ψ 2 ψ n ψn ψn 2 ψ3 ψ2 ψ (22 7

8 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 where ψ = K2 F 0 ( F n 3 F 0 ψ 2 = F 0 K2 F 0 ψ 3 = K2 F 0 ψ 4 = K2 F 0 a + F n 4 F 0 n 4 a 2 + n 4 (xa + ya 2 n 4 a n 2 F k F 0 Fk a n 2 kn 2 n 4 (a n 3 + a n 2 F k F 0 ψ j = K2 (a n+ j + a n+2 j a n+3 j n 4 ( F n 3 F 0 a n 2 k + F n 4 a n 2 k2 (xa n 2 k + ya n 2 k2 Fk (a n 2 kn 3 + a n 2 kn 2 Fk (a n 2 kn+ j + a n 2 kn+2 j a n 2 kn+3 j (5 j n ψ n = K2 F 0 ψ 22 = K F 0 ψ 23 = K F 0 ψ 24 = K F 0 n 4 (a a 3 n 3 Fk (xa + ya 2 n 3 a n 2 Fk (a n 2 k a n 2 k3 Fk a n kn 2 n 3 (a n 3 + a n 2 (xa n k + ya n k2 Fk (a n kn 3 + a n kn 2 ψ 2j = K (a n+ j + a n+2 j a n+3 j n 3 Fk (a n kn+ j + a n kn+2 j a n kn+3 j (5 j n ψ 33 = n 3a n 2 + a n 2n 2 ψ i4 = n i(a n 3 + a n 2 + (a n i+n 3 + a n i+n 2 (3 i 4 ψ ij = n i(a n+ j + a n+2 j a n+3 j + (a n i+n+ j + a n i+n+2 j a n i+n+3 j in which a ij is defined by ( a = n 5 ω n 3 ω n 4b α s (3 i [ n + ]; i j n + i; except ψ 33 ψ 34 and ψ 44 2 n 4 k ω k (b n 3 k α + b jt n 4 k j 8

9 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 ( a 2 = n 5 n 4 k β n 3 β n 4b α β k (b n 3 k α + b jt n 4 k j a 2 = d s s a22 = c s a ij = n j b k (a iβ n j k + a i2ω n j k ( i 2 3 j n 2 b αa 2j (i = 3 j 2 i 3 a 2j(b i 2α + b k t i 3 k (4 i n 2 j 2 ( n 4 b α b k (a 2β n 3 k + a 22ω n 3 k + b (i = 3 j = 3 ( n j b α b k (a 2β n j k + a 22ω n j k (i = 3 4 j n 2 ( i 3 n j (b i 2α + b k t i 3 k b k (a 2β n j k + a 22ω n j k + b i j+ i 3 (b i 2α + b k t i 3 k ( n j (4 i n 2 3 j n 2 i j b k (a 2β n j k + a 22ω n j k (4 i n 2 3 j n 2 i < j with ( n 5 n 4 k s = c ω n 3 ω n 4b α ω k (b n 3 k α + b jt n 4 k j ( n 5 n 4 k d β n 3 β n 4b α β k (b n 3 k α + b jt n 4 k j b i = det p i ( i n 4 c = F n 3 F 0 β i = F n 3 F i + ξ i ω i = F n 4 F i + η i ( i n 3 K + K 3 d = F n 4 K + K 4 K K 2 K 3 K 4 x y α i(0 i n 3 ξ i( i n 3 η i( i n 3 and t i(0 i n 6 are as in Theorem 2 det p i(0 i n 5 is the determinant of Hessenberg matrix and can be calculated by Lemma [x] = m m x < m + m is an integer We can observe that T Fn is not only a Hermitian matrix but also a symmetric matrix along its secondary diagonal ie sub-symmetric matrix 9

10 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 Proof We can introduce the following two transformation matrices when n > 6 M 2 = F n 3 F 0 0 F n 4 F N 2 = 0 K2 F 0 F n 4 F 0 F n 5 F 0 F F 0 F 0 F 0 0 K F 0 F n 3 F 0 F n 4 F 0 F 2 F 0 F F If we multiply M T F n N by M 2 and N 2 the M and N are as in the proof of Theorem 2 so we obtain M 2M T F n N N 2 = 0 F F c β n 3 β n 4 β n 5 β 4 β 3 β 2 β d ω n 3 ω n 4 ω n 5 ω 4 ω 3 ω 2 ω 0 α 0 0 t 0 α t t 0 α t n 6 t t 0 α 0

11 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 with c = F n 3 F 0 d = F n 4 F 0 K + K 3 β i = F n 3 F i + ξ i ( i n 3 K + K 4 ω i = F n 4 F i + η i ( i n 3 We have M 2M T F n N N 2 = Φ Λ where Φ = ( 0 F and Λ is a Toeplitz-like matrix Λ = c β n 3 β n 4 β n 5 β 4 β 3 β 2 β d ω n 3 ω n 4 ω n 5 ω 4 ω 3 ω 2 ω 0 α 0 0 t 0 α t t 0 α t n 6 t t 0 α (n 2 (n 2 Φ Λ is the direct sum of Φ and Λ Let M = M 2M N = N N 2 then we obtain T Fn = N (Φ Λ M M = x F n 3 F 0 F n 4 F 0 y

12 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 N = 0 K2 0 K F 0 F n 4 F n 3 F n 5 F n 4 F 2 F 3 F F 2 F 0 F 0 n n 4 0 n We can observe that the inverse matrix of Φ is Φ = Let Λ be partitioned as Λ = ( J V U B c β n 3 β n 4 β n 5 β 4 β 3 β 2 β 0 α t 0 α t t 0 α = t n 4 α t0 α 0 0 t n 6 t n 7 t t 0 α d ω n 3 ω n 4 ω n 5 ω 4 ω 3 ω 2 ω } {{ } n 4 0 F 0 F 0 0 According to Lemma in [7] we have B = b 0 0 b 2 b b 3 b 2 b 0 b n 4 b 3 b 2 b where b i = det p i ( i n 4 det p i(0 i n 5 is the determinant of Hessenberg matrix and can be calculated by Lemma 2

13 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 Suppose that T F n is invertible and so is Λ Since B is invertible the Schur complement of B denoted by S is also invertible and we have S = J V B U ( c βn 3 = d c = d ω n 3 ( βn 4 β n 5 β ω n 4 ω n 5 ω b 0 0 b 2 b 0 b n 4 b 2 b n 5 n 4 k β n 3 β n 4b α β k (b n 3 k α + b jt n 4 k j n 5 ω n 3 ω n 4b α ω k (b n 3 k α + n 4 k b jt n 4 k j 0 α 0 t 0 t t 2 0 t n 6 with the determinant: ( n 5 n 4 k s = det S = c ω n 3 ω n 4b α ω k (b n 3 k α + b jt n 4 k j ( n 5 n 4 k d β n 3 β n 4b α β k (b n 3 k α + b jt n 4 k j Hence the inverse of Λ is given with the following form: ( Λ = [a ij] ij n 2 = S B US S V B B US V B + B where [a ij] ij n 2 are the same as in Theorem 22 We obtain that Φ Λ = 0 F a a 2 a 3 a n 3 a n 2 a 2 a 22 a 23 a 2n 3 a 2n 2 a 3 a 32 a 33 a 3n 3 a 3n 2 a n 3 a n 32 a n 33 a n 3n 3 a n 3n a n 2 a n 22 a n 23 a n 2n 3 a n 2n 2 3

14 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 and we have T F n = N (Φ Λ M = ψ ψ 2 ψ 3 ψ n 2 ψ n ψ n ψ 2 ψ 22 ψ 23 ψ 2n 2 ψ 2n ψ n ψ 3 ψ23 ψ 33 ψ 3n 3 ψ 2n 2 ψ n 2 ψ n 2 ψ2n 2 ψ3n 3 ψ 33 ψ 23 ψ 3 ψ n ψ2n ψ2n 2 ψ23 ψ 22 ψ 2 ψ n ψn ψn 2 ψ3 ψ2 ψ where [ψ ij]( i [ n+ ]; i j n + j are the same as in Theorem 22 2 Which completes the proof 3 Numerical Example In this section an example demonstrates the method which introduced above for the calculation of determinant and inverse of the complex Fibonacci Hermitian Toeplitz matrix Here we consider a 7 7 matrix: T F 7 = 0 i + i + 2i 2 + 3i 3 + 5i 5 + 8i i 0 i + i + 2i 2 + 3i 3 + 5i i i 0 i + i + 2i 2 + 3i 2i i i 0 i + i + 2i 2 3i 2i i i 0 i + i 3 5i 2 3i 2i i i 0 i 5 8i 3 5i 2 3i 2i i i 0 Using the corresponding formulas in Theorem 2 we get x = 8 5i y = 5 3i = + 2i α = 2i t 0 = + 2i t = 2 + 2i K = 30 2i K 3 = i K 4 = + 08i and from Lemma we can obtain det 4([η k ] 4 α t 0 t = 0 40i det 4([ξ k ] 4 α t 0 t = 5 64i det 4([F k ] 4 α t 0 t = 4 + 3i 7 7 From (2 we obtain det T F 7 = [ (K 3 det 4([η k ] 4 α t 0 t K 4 det 4([ξ k ] 4 α t 0 t F 4 (K det 4([η k ] 4 α t 0 t K 4 det 4([Fk ] 4 α t 0 t + F 3 (K det 4([ξ k ] 4 α t 0 t K 3 det 4([Fk ] 4 α t 0 t ] = 32 4

15 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 As the inverse calculation if we use the corresponding formulas in Theorem 22 we get so we can get T F 7 = 3 32 ψ = 3 32 ψ2 = 3 25 i ψ3 = 9 i ψ4 = i 8 32 ψ 5 = 9 + i 32 6 ψ6 = i ψ7 = i ψ22 = 3 32 ψ 23 = i ψ24 = 3 5 i ψ25 = i ψ26 = i ψ 33 = 9 32 ψ34 = 9 7 i 6 32 ψ35 = 5 6 i ψ44 = i 25 9 i i 9 + i 7 3 i 3 i i i 3 3 i 5 7 i 35 + i 7 3 i i 7 7 i i 5 i 5 7 i 9 + i i i i i 3 3 i i i i i i i i i i i i i i i i 9 i 5 29 i i + 3 i Conclusion In this paper by constructing the special transformation matrices we give the determinant and inverse of the complex Fibonacci Hermitian Toeplitz matrix in section 2 Acknowledgement This project is supported by the NSFC (Grant No Competing Interests Authors have declared that no competing interests exist References [] Horadam AF Complex Fibonacci numbers and Fibonacci quaternions Amer Math Monthly 963;70: [2] Mukhexjee BN Maiti SS On some properties of positive definite Toeplitz matrices and their possible applications Linear Algebra Appl 988;02:2-240 [3] Grenander U Szego G Toeplitz forms and their applications Univ of California Press Berkeley; 958 [4] Heining G Rost K Algebraic methods for Toeplitz-Like Matrices and operators Akademie- Verlag Berlin; 984 5

16 Sun and Jiang; BJMCS 9( ; Article nobjmcs30398 [5] Lin D Fibonacci-Lucas quasi-cyclic matrices Fibonacci Quart 2002;40: [6] Shen SQ Cen JM Hao Y On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers Appl Math Comput 20;27: [7] Bozkurt D Tam TY Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas Numbers Appl Math Comput 202;29: [8] Jiang ZL Gong YP Gao Y Invertibility and explicit inverses of circulant-type matrices with k-fibonacci and k-lucas numbers Abstr Appl Anal 204; 0 Article ID [9] Li J Jiang ZL Lu FL Determinants norms and the spread of circulant matrices with Tribonacci and generalized Lucas numbers Abstr Appl Anal 204;9:204 Article ID [0] Jiang XY Hong KC Exact determinants of some special circulant matrices involving four kinds of famous numbers Abstr Appl Anal 204;2 Article ID [] Jiang ZL Gong YP Gao Y Circulant type matrices with the sum and product of Fibonacci and Lucas numbers Abstr Appl Anal 204;2 Article ID [2] Jiang ZL Zhou JW A note on spectral norms of even-order r-circulant matrices Appl Math Comput 205;250: [3] Zhou JW Jiang ZL The spectral norms of g-circulant matrices with classical Fibonacci and Lucas numbers entries Appl Math Comput 204;233: [4] Jiang XY Hong KC Explicit inverse matrices of Tribonacci skew circulant type matrices Appl Math Comput 205;268:93-02 [5] Zheng YP Shon S Exact determinants and inverses of generalized Lucas skew circulant type matrices Appl Math Comput 205;270:05-3 [6] Liu L Jiang ZL Explicit form of the inverse matrices of Tribonacci circulant type matrices Abstr Appl Anal 205;0 Article ID [7] Xiaoting Chen Zhaolin Jiang Jianmin Wang Determinants and Inverses of Fibonacci and Lucas Skew Symmetric Toeplitz Matrices British Journal of Mathematics & Computer Science 206;9(2:-2 Article nobjmcs28848 c 206 Sun and Jiang; This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Peer-review history: The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar 6