Determinant and inverse of a Gaussian Fibonacci skew-hermitian Toeplitz matrix

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1 Available online at wwwisr-publicationscom/nsa J Nonlinear Sci Appl 1 (17) Research Article Journal Homepage: wwwtnsacom - wwwisr-publicationscom/nsa Determinant and inverse of a Gaussian Fibonacci skew-hermitian Toeplitz matrix Zhaolin Jiang a Jixiu Sun ab a Department of Mathematics Linyi University Linyi 76 P R China b School of Mathematics and Statistics Shandong Normal University Jinan 514 P R China Communicated by S-M Jung Abstract In this paper we consider the determinant and the inverse of the Gaussian Fibonacci skew-hermitian Toeplitz matrix We first give the definition of the Gaussian Fibonacci skew-hermitian Toeplitz matrix Then we compute the determinant and inverse of the Gaussian Fibonacci skew-hermitian Toeplitz matrix by constructing the transformation matrices c 17 All rights reserved Keywords: Gaussian Fibonacci number skew-hermitian Toeplitz matrix determinant inverse 1 MSC: 15B5 15A9 15A15 1 Introduction The Gaussian Fibonacci sequence [9 1] is defined by the following recurrence relations: G n+1 = G n + G n 1 n 1 with the initial condition G = i G 1 = 1 The {G n } is given by the formula G n = (1 iβ)αn + (iα 1)β n α β = αn β n + (α n 1 β n 1 )i α β α and β are the roots of the characteristic equation x x 1 = We can observe that Ḡ n+1 = Ḡ n + Ḡ n 1 n 1 and G n = F n + if n 1 F n is Fibonacci number [1] On the other hand skew-hermitian Toeplitz matrix [3 4 1] and Hermitian Toeplitz matrix have important applications in various disciplines including image processing signal processing and solving least squares problems [7 8 ] It is an ideal research area and hot topic for the inverses of the Toeplitz matrices [15 5] Some scholars showed the explicit determinants and inverses of the special matrices involving famous numbers Corresponding author addresses: zh18@sinacom (Zhaolin Jiang) sunixiu1314@163com (Jixiu Sun) doi:1436/nsa177 Received

2 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) Dazheng showed the determinant of the Fibonacci-Lucas quasi-cyclic matrices in [6] Circulant matrices with Fibonacci and Lucas numbers are discussed and their explicit determinants and inverses are proposed in [] The authors provided determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers in [] 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 [17] In [1] 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 Jiang et al [11] 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 In [13] 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 It should be noted that Jiang and Zhou [16] 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 [6] investigated explicit formulas of spectral norms for g-circulant matrices with Fibonacci and Lucas numbers Furthermore in [14] the determinants and inverses are discussed and evaluated for Tribonacci skew circulant type matrices The authors [4] 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 [18] Sun and Jiang [3] gave the determinant and inverse of the complex Fibonacci Hermitian Toeplitz matrix by constructing the transformation matrices Determinants and inverses of Fibonacci and Lucas skew symmetric Toeplitz matrices are given by constructing the special transformation matrices in [5] The purpose of this paper is to obtain better results for the determinant and inverse of Gaussian Fibonacci skew-hermitian Toeplitz type matrix In this paper we adopt the following two conventions = 1 i = 1 and we define a kind of special matrix as follows Definition 11 A Gaussian Fibonacci skew-hermitian Toeplitz matrix is a square matrix of the form G G 1 G G n 3 G n G n 1 Ḡ 1 G G 1 Gn 3 G n Ḡ Ḡ 1 G Gn 3 T Gn = Ḡ n 3 G G 1 G Ḡ n Ḡ n 3 Ḡ1 G G 1 Ḡ n 1 Ḡ n Ḡ n 3 Ḡ Ḡ 1 G G G 1 G n 1 are the Gaussian Fibonacci numbers It is evidently determined by its first row Preliminaries The determinant and the inverse of some special structured matrices are discussed in this section These matrices can be met in course of Gaussian Fibonacci skew-hermitian Toeplitz matrix determinantcalculation (or inverse-calculation) Lemma 1 ([19]) A Toeplitz-Hessenberg matrix is an n n matrix of the form κ 1 κ κ κ 1 κ T H = κ 4 κ κ 1 κ κ κ n 1 κ κ 1 κ κ n κ κ 1

3 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) κ i for at least one i > Let n be a positive integer Then the determinant of an n n Toeplitz-Hessenberg matrix is t1 + + t det T H = n ( κ t 1 t ) n t 1 t n 1 κt κt n n n t 1 +t + +nt n =n ( t1 + + t n t 1 t n ) = (t t n )! t 1! t n! is the multinomial coefficient and n = t 1 + t + + nt n is a partition of the positive integer n each positive integer i appears t i times Lemma Define an i i lower-triangular Toeplitz-like matrix by µ i µ i 1 µ i µ i 3 µ i 4 µ µ 1 κ 1 κ κ κ 1 κ i ([µ k ] i k=1 κ κ 3 κ κ 1 κ κ 1 κ i 1 ) = κ 4 κ 3 κ κ 1 κ κ3 κ κ 1 κ κ i 1 κ 4 κ 3 κ κ 1 κ we have i i k det 1 (µ 1 ) = µ 1 det i ([µ k ] i k=1 κ κ 1 κ i 1 ) = µ i κ i 1 i 1 + =1 t1 + + t t 1 t ( 1) + µ i κ i 1 = (t t )! t 1! t! t 1 +t + +t = ( t1 + + t t 1 t ) ( κ ) t 1 t 1 κt i Proof By using the Laplace expansion of matrix i ([µ k ] i k=1 κ κ 1 κ i 1 ) (i ) along the first row and Lemma 1 it is easy to check that det i ([µ k ] i k=1 κ κ 1 κ i 1 ) = ( 1) 1+1 µ i κ i 1 + ( 1) 1+ µ i 1 κ i + ( 1) 1+3 µ i κ i 3 t1=1 ( t1 t 1 +t = t 1 ) ( κ ) 1 t 1 1 ( t1 + t t 1 t ) ( κ ) t 1 t 1 κt + ( 1) 1+i µ 1 κ t 1 +t + +(i 1)t i 1 =i 1 ( t1 + + t i 1 t 1 t i 1 ) ( κ ) i 1 t 1 t i 1 1 κt i 1 i 1

4 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) = µ i κ i 1 i 1 + =1 t1 + + t t 1 t ( 1) + µ i κ i 1 = (t t )! t 1! t! t 1 +t + +t = t1 + + t ( κ t 1 t ) t 1 t 1 κt Lemma 3 Define an i i lower-triangular Toeplitz-like matrix by J i ([µ k ] i k=1 κ κ 1 κ i ) = µ i µ i 1 µ i µ i 3 µ i 4 µ µ 1 κ κ 1 κ κ κ 1 κ κ 3 κ κ 1 κ κ i 3 κ κ 1 κ κ i κ κ 1 κ i i with µ 1 κ then its inverse J 1 i ([µ k ] i k=1 κ κ 1 κ i ) is given by b 1 b b 1 b 3 b b 1 J 1 i ([µ k ] i k=1 κ κ 1 κ i ) = b i 1 b i b 3 b b 1 B 1 B B 3 B i B i 1 B i i i b 1 = 1 κ b = ( 1) 1 κ i 1 t 1 +t + +( 1)t 1 = 1 ( t1 + + t 1 t 1 t 1 ) ( κ ) ( 1) t 1 t 1 1 κt 1 1 with B 1 = 1 µ 1 B = ( 1) 1 det i+1 ([µ k ] i+ κ κ 1 κ i ) µ 1 κ i+1 ( i) det 1 (µ ) = µ det i ([µ k ] i+1 κ κ 1 κ i 1 ) = µ i+1 κ i 1 i 1 + =1 ( 1) + µ i +1 κ i 1 t 1 +t + +t = t1 + + t ( κ t 1 t ) t 1 t 1 κt i

5 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) t1 + + t and t 1 t = (t t )! t 1! t! Proof By using the Laplace expansion of matrix J i ([µ k ] i k=1 κ κ 1 κ i ) along the last column it is easy to check that det J i ([µ k ] i k=1 κ κ 1 κ i ) = ( 1) 1+i µ 1 κ i 1 according to the sufficient and necessary conditions for the invertibility of matrices we obtain that J i ([µ k ] i k=1 κ κ 1 κ i ) is invertible and the augmented matrix of J i ([µ k ] i k=1 κ κ 1 κ i ) is as follows b 1 b b 1 b 3 b b 1 J i ([µ k] i k=1 κ κ 1 κ i ) = b i 1 b i b 3 b b 1 B 1 B B 3 B i B i 1 B i b 1 = ( 1)i+1 µ 1 κ i b = ( 1)i+ µ 1 κ i 1 i 1 B 1 = ( 1)1+i κ i 1 t 1 +t + +( 1)t 1 = 1 B = ( 1)+i κ det i+1 ([µ k ] i+ κ κ 1 κ i ) i with det 1 (µ ) = µ det i ([µ k ] i+1 κ κ 1 κ i 1 ) = µ i+1 κ i 1 i 1 + =1 ( 1) + µ i +1 κ i 1 t 1 +t + +t = i i t1 + + t 1 ( κ t 1 t ) ( 1) t 1 t 1 1 κt t1 + + t ( κ t 1 t ) t 1 t 1 κt i t1 + + t and = (t t )! t 1 t t 1! t! Then J 1 i ([µ k ] i k=1 κ 1 κ 1 κ i ) = ( 1) 1+i µ 1 κ i 1 J i ([µ k] i k=1 κ κ 1 κ i ) and we obtain the inverse of J i ([µ k ] i k=1 κ κ 1 κ i ) as described in Lemma 3 3 Determinant and inverse of the Gaussian Fibonacci skew-hermitian Toeplitz matrix Let T Gn be an invertible Gaussian Fibonacci skew-hermitian Toeplitz matrix In this section we give the determinant and inverse of the matrix T Gn

6 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) Theorem 31 Let T Gn be a Gaussian Fibonacci skew-hermitian Toeplitz matrix then we have and det T G1 = i det T G = det T G3 = 5i det T G4 = 5 det T Gn = ( 1) n 1 ( + i) n 3 G {(Ḡn G 1 n + G Ḡ n 3 )[Ḡn 1G n 1 G + G n 1 k (Ḡn 1G k Ḡ n 1 k )] G (Ḡn 1G 1 Ḡ n G )[Ḡn G n 1 + G 1 G G n i (Ḡn G n + G ) + n 1 k G (Ḡn G k Ḡ n k )]} n > 4 G = 1 1 = G 1 + i i = + i (G i 1 i + G k i k ) ( i n 3) Proof Let T Gn be a Gaussian Fibonacci skew-hermitian Toeplitz matrix and we can easily get the following conclusions: and k=1 det T G1 = i det T G = det T G3 = 5i det T G4 = 5 We can introduce the following two transformation matrices when n > 4 1 x 1 y 1 M 1 = n 3 1 N 1 = n x = Ḡn 1 y = Ḡn = 1 1 = G 1 G G + i i = + i (G i + i 1 G k i k ) ( i n 3) k=1 (31)

7 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) By using M 1 N 1 and the recurrence relations of the Gaussian Fibonacci sequences the matrix T Gn is changed into the following form G K 1 G n G n 3 G 4 G 3 G G 1 ξ n ξ n 3 ξ 4 ξ 3 ξ ξ 1 K 3 η n η n 3 η 4 η 3 η η 1 α M 1 T Gn N 1 = G 1 α G G 1 G n 4 G G 1 α while K 1 = n 1 G k n 1 k = n 1 ξ i = xg i Ḡ n i 1 (1 i n ) ξ n 1 = xg n 1 + G ξ k n 1 k K 3 = n 1 η k n 1 k η i = yg i Ḡ n i (1 i n 3) η n = yg n + G η n 1 = yg n 1 + G 1 α = + i By using the Laplace expansion of matrix M 1 T Gn N 1 along the first column we can get that det(m 1 T Gn N 1 ) = ( 1) n 1 α n 3 G ( η 1 K 3 ξ 1 ) = ( 1) n 1 ( + i) n 3 G {(Ḡn G 1 n + G Ḡ n 3 )[Ḡn 1G n 1 G + G n 1 k (Ḡn 1G k Ḡ n 1 k )] G (Ḡn 1G 1 Ḡ n G )[Ḡn G n 1 + G 1 G G n i (Ḡn G n + G ) + n 1 k G (Ḡn G k Ḡ n k )]} G det M 1 = det N 1 = ( 1) (n 1)(n ) we can obtain det T Gn as (31) which completes the proof Theorem 3 Let T Gn be an invertible Gaussian Fibonacci skew-hermitian Toeplitz matrix and n > 5 Then we have ψ 11 ψ 1 ψ 13 ψ 1n ψ 1n 1 ψ 1n ψ 1 ψ ψ 3 ψn 1 ψ 1n 1 ψ 13 ψ 3 ψ 33 ψ1n T 1 Gn = ψ 1n ψ33 ψ 3 ψ 13 ψ 1n 1 ψ n 1 ψ 3 ψ ψ 1 ψ 1n ψ 1n 1 ψ 1n ψ 13 ψ 1 ψ 11

8 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) ψ 11 = 1 K 1x + D 1 B 1 ( K 3x + y) G G ψ 1 = D b 1 + D 1 B n ψ 13 = D 3 b 1 + D (b + b 1 ) + D 1 (B n 3 + B n ) ψ 1 = D b 1 + D 1 (b + b 1 ) + D 1 (B n + B n +1 B n + ) + D k (b k+1 + b k b k 1 ) (4 n ) n 3 ψ 1n 1 = D n b 1 + D 1 (B 1 + B B 3 ) + D k (b n 1 k b n k ) in which ψ 1n = K 1 D 1 ( K n 3B 1 + B ) D k b n 1 k G ψ = B n ψ 3 = B n 3 + B n ψ = B n + B n +1 B n + (4 n 1) ψ 33 = a n 33 b 1 + a n 3 (b + b 1 ) + a n 31 (B n 3 + B n ) ψ i = a n i b 1 + a n i 1 (b + b 1 ) + a n i1 (B n + B n +1 B n + ) [ ] n a n ik (b k+1 + b k b k 1 ) (3 i ; i n + 1 i; except ψ 33 ) D i = K 1ξ i G i (1 i n ) ω i = K 3 ξ i + η i (1 i n ) G G ξ i i + n 1 a i = ξ ( i n 3 1 n ) i + 1 i + = n 1 b 1 = 1 α b i = ( 1) i 1 α i ( i n 3) B 1 = 1 ω 1 t 1 +t + +(i 1)t i 1 =i 1 t1 + + t i 1 ( α) (i 1) t 1 t i 1 G t 1 t 1 t 1 Gt i 1 i 1 i 1 B i = ( 1)i 1 det n 1 i ([ω k ] n i α G 1 G G n i ) ω 1 α n 1 i ( i n ) det 1 (ω ) = ω det i ([ω k ] i+1 α G i 1 1 G i 1 ) = ω i+1 α i 1 + ( 1) + ω i +1 α i 1 t1 + + t with t 1 t t 1 +t + +t = =1 t1 + + t ( α) t 1 t G t 1 t 1 t 1 Gt i n 3 = (t t )! x y α K 1 K 3 i ( i n 3) η i (1 i n 1) ξ i (1 t 1! t!

9 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) i n 1) as in Theorem 31 [x] = m m x < m + 1 m is an integer We can observe that T 1 Gn is not only a skew-hermitian matrix but also a symmetric matrix along its secondary diagonal ie sub-symmetric matrix Proof We can introduce the following two transformation matrices when n > K 3 1 M = N = 1 K 1 G D n D n 3 D D 1 1 ξ n ξ n 3 ξ ξ D i = K 1ξ i G i (1 i n ) G G If we multiply M 1 T n N 1 by M and N the M 1 and N 1 are as in the proof of Theorem 31 so we obtain G ω n ω n 3 ω 4 ω 3 ω ω 1 α M M 1 T n N 1 N = G 1 α G G 1 G n 4 G G 1 α ω i = K 3 ξ i + η i (1 i n ) We have M M 1 T n N 1 N = Φ Λ

10 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) G Φ = is a diagonal matrix and Λ is a lower-triangular Toeplitz-like matrix ω n ω n 3 ω n 4 ω n 5 ω 4 ω 3 ω ω 1 α G 1 α G G 1 α Λ = G G 1 α G1 α G n 4 G G 1 α Φ Λ is the direct sum of Φ and Λ Let M = M M 1 N = N 1 N then we obtain (n ) (n ) T 1 Gn = N(Φ 1 Λ 1 )M 1 x 1 K 3 x + y 1 K 3 M = K 1 D n D n 3 D D 1 G 1 n 3 a n 3n a n 3n 3 a n 3 a n 31 N = n 4 a n 4n a n 4n 3 a n 4 a n 41 1 a 1n a 1n 3 a 1 a 11 1 a n a n 3 a a 1 a i = ξ i i + n 1 ξ i + 1 i + = n 1 We can observe that the inverse matrix of Φ is ( i n 3 1 n ) 1 G 1

11 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) According to Lemma 3 we have b 1 b b 1 Λ 1 b 3 b = b b 1 b n 3 b 3 b b 1 B 1 B B 3 B n 4 B n 3 B n (n ) (n ) b 1 = 1 α b i = ( 1) i 1 α i ( i n 3) t 1 +t + +(i 1)t i 1 =i 1 ( t1 + + t i 1 t 1 t i 1 ) ( α) (i 1) t 1 t i 1 G t 1 1 Gt i 1 i 1 B 1 = 1 ω 1 B i = ( 1)i 1 det n 1 i ([ω k ] n i α G 1 G G n i ) ω 1 α n 1 i ( i n ) in which det 1 (ω ) = ω det i ([ω k ] i+1 α G 1 G i 1 ) i 1 = ω i+1 α i 1 + ( 1) + ω i +1 α i 1 t1 + + t and t 1 t We obtain that =1 t 1 +t + +t = = (t t )! t 1! t! Φ 1 Λ 1 = t1 + + t ( α) t 1 t G t 1 t 1 t 1 Gt ( i n 3) 1 G 1 b 1 b b 1 b b n 3 b b 1 B 1 B B 3 B n 3 B n

12 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) then we have T 1 Gn = N(Φ 1 Λ 1 )M ψ 11 ψ 1 ψ 13 ψ 1n ψ 1n 1 ψ 1n ψ 1 ψ ψ 3 ψn 1 ψ 1n 1 ψ 13 ψ 3 ψ 33 ψ1n = ψ 1n ψ33 ψ 3 ψ 13 ψ 1n 1 ψ n 1 ψ 3 ψ ψ 1 ψ 1n ψ 1n 1 ψ 1n ψ 13 ψ 1 ψ 11 [ψ i ](1 i [ n+1 ]; i n + 1 i) are the same as in Theorem 3 which completes the proof 4 Numerical example In this section an example demonstrates the method which introduced above for the calculation of determinant and inverse of the Gaussian Fibonacci skew-hermitian Toeplitz matrix Here we consider a 6 6 matrix: T G6 = i i + i 3 + i 5 + 3i 1 i i + i 3 + i 1 + i 1 i i + i + i 1 + i 1 i i 3 + i + i 1 + i 1 i i 3 + i + i 1 + i 1 i Using the corresponding formulas in Theorem 31 we get 6 6 From (31) we obtain 1 = i = i 3 = i det T G6 = ( 1) 5 ( + i) 3 G {(Ḡ4G 1 Ḡ 3 G )[Ḡ5G 5 + G G k (Ḡ5G k Ḡ 5 k )] G (Ḡ5G 1 Ḡ 4 G )[Ḡ4G 5 + G 1 G + 1 (Ḡ4G 4 G + G ) k (Ḡ4G k G Ḡ 4 k )]} = 45 As the inverse calculation if we use the corresponding formulas in Theorem 3 we get ψ 11 = 4 9 i ψ 1 = i ψ 13 = 9 ψ 14 = i ψ 15 = i ψ 16 = i ψ = 5 9 i ψ 3 = i ψ 4 = i ψ 5 = i ψ 33 = 1 9 i ψ 34 = i

13 Z Jiang J Sun J Nonlinear Sci Appl 1 (17) so we can get T 1 G6 = 4 9 i i i i i i 5 9 i i i i i i 1 9 i i i i i i i 1 9 i i i i i i 5 9 i i i i i i 4 9 i Conclusion We deal with two kinds of lower triangular Toeplitz-like matrices base on the determinant of the Toeplitz-Hessenberg matrix the determinants and inverses of such special matrices are presented respectively By constructing the special transformation matrices and using the determinant of the lower triangular Toeplitz-like matrix we obtain the determinant and inverse of a Gaussian Fibonacci skew-hermitian Toeplitz matrix Acknowledgment The research was supported by National Natural Science Foundation of China (Grant No ) Natural Science Foundation of Shandong Province (Grant No ZR16AM14) and the AMEP of Linyi University China References [1] M Akbulak D Bozkurt On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers Hacet J Math Stat 37 (8) [] D Bozkurt T-Y Tam Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas Numbers Appl Math Comput 19 (1) [3] A Buckley On the solution of certain skew symmetric linear systems SIAM J Numer Anal 14 (1977) [4] R H Chan X-Q Jin Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems BIT 31 (1991) [5] X-T Chen Z-L Jiang J-M Wang Determinants and inverses of Fibonacci and Lucas skew symmetric Toeplitz matrices British J Math Comput Sci 19 (16) [6] L Dazheng Fibonacci-Lucas quasi-cyclic matrices A special tribute to Calvin T Long Fibonacci Quart 4 () [7] U Grenander G Szegö Toeplitz forms and their applications California Monographs in Mathematical Sciences University of California Press Berkeley-Los Angeles (1958) 1 [8] G Heining K Rost Algebraic methods for Toeplitz-like matrices and operators Mathematical Research Akademie- Verlag Berlin (1984) 1 [9] A F Horadam Further appearence of the Fibonacci sequence Fibonacci Quart 1 (1963) [1] A İpek K Arı On Hessenberg and pentadiagonal determinants related with Fibonacci and Fibonacci-like numbers Appl Math Comput 9 (14) [11] Z-L Jiang Y-P Gong Y Gao Circulant type matrices with the sum and product of Fibonacci and Lucas numbers Abstr Appl Anal 14 (14) 1 pages 1 [1] Z-L Jiang Y-P Gong Y Gao Invertibility and explicit inverses of circulant-type matrices with k-fibonacci and k-lucas numbers Abstr Appl Anal 14 (14) 1 pages 1 [13] X-Y Jiang K-C Hong Exact determinants of some special circulant matrices involving four kinds of famous numbers Abstr Appl Anal 14 (14) 1 pages 1

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