Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 483021, 10 pages http://dx.doi.org/10.1155/2014/483021
Research Article On Skew Circulant Type Matrices Involving Any Continuous Fibonacci Numbers
Zhaolin Jiang, Jinjiang Yao, and Fuliang Lu
School of Science, Linyi University, Shuangling Road, Linyi 276005, China
Correspondence should be addressed to Jinjiang Yao; [email protected]
Received 9 April 2014; Accepted 30 April 2014; Published 18 June 2014
Academic Editor: Zidong Wang
Copyright © 2014 Zhaolin Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Circulant and skew circulant matrices have become an important tool in networks engineering. In this paper, we consider skew circulant type matrices with any continuous Fibonacci numbers. We discuss the invertibility of the skew circulant type matrices and present explicit determinants and inverse matrices of them by constructing the transformation matrices. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm and bounds for the spread of these matrices are given, respectively.
1. Introduction The system model of the OFDM based AF relay networks as well as the strategy of the superimposed training involves Skew circulant and circulant matrices have important appli- circulant matrix [10]. Two-way transmission model consid- cations in various networks engineering. Joy and Tavsanoglu ered in [11] ensured the circular convolution between two [1] showed that feedback matrices of ring cellular neural frequency selective channels. networks,whichcanbedescribedbytheODE,areblock The skew circulant matrices as preconditioners for linear circulants. A special class of the feedback delay network using multistep formulae- (LMF-) based ordinary differential equa- circulant matrices was proposed [2]. Jing and Jafarkhani tions (ODEs) codes, Hermitian, and skew-Hermitian Toeplitz [3] proposed distributed differential space-time codes that systems were considered in [12–15]. Lyness and Sørevik work for networks with any number of relays using circulant employed a skew circulant matrix to construct s-dimensional matrices. Exploiting the circulant structure of the channel lattice rules in [16]. Compared with cyclic convolution matrices, Eghbali et al. [4]analysedtherealisticnearfast algorithm, the skew cyclic convolution algorithm [17]was fading scenarios with circulant frequency selective channels. able to perform filtering procedure in approximately half of Rocchesso [5] presented particular choices of the feedback computational cost for real signals. In [18]twonewnormal- coefficients, namely, Galois sequences, arranged in a circulant form realizations were presented which utilize circulant and matrix, to produce a maximum echo density in the time skew circulant matrices as their state transition matrices. The response. Sardellitti et al. [6] used an analytical expression well-known second-order coupled form is a special case of for the eigenvalues of a block circulant matrix as a function the skew circulant form. Li et al. [19]gavethestylespectral of the coverage radius. Li et al. [7] gave a low-complexity decomposition of skew circulant matrix firstly and then dealt binary frame-wise network coding encoder design based on with the optimal backward perturbation analysis for the circulant matrix. Hirt and Massey [8] introduced discrete linear system with skew circulant coefficient matrix. In [20], time Fourier transform precoding to the proposed multihop a new fast algorithm for optimal design of block digital filters relay system involving circulant matrix. When considering (BDFs) was proposed based on skew circulant matrix. a single-input single-output transmission with CFO and Besides, some scholars have given various algorithms omitting the relay index subscript, Wang et al. [9]proved for the determinants and inverses of nonsingular circulant that the intercarrier interference matrix is a circulant matrix. matrices. Unfortunately, the computational complexity of 2 Abstract and Applied Analysis these algorithms is very amazing huge with the order of to physics of high energy particles [40].Asisshownin[41, matrix increasing. However, some authors gave the explicit 42], the hyperbolic Fibonacci functions can lead to creation of determinants and inverses of circulant and skew circulant the Lobachevsky Fibonacci and Minkovsky Fibonacci geom- matrices involving some famous numbers. For example, Yao etry which are of great importance for theoretical physics. In and Jiang [21] considered the determinants, inverses, norm, the 19th century the French mathematician Francois Edouard and spread of skew circulant type matrices involving any Anatole Lucas (1842–1891) introduced the so-called Lucas continuous Lucas numbers. Shen et al. considered circulant numbers given by the recursive relation 𝐿𝑛 =𝐿𝑛−1 +𝐿𝑛−2,𝑛≥ matrices with Fibonacci and Lucas numbers and presented 2, with the seeds 𝐿0 =2and 𝐿1 =1. The determinants, their explicit determinants and inverses by constructing the inverses, norm, and spread of skew circulant type matrices transformation matrices [22]. Gao et al. [23] gave explicit involving any continuous Lucas numbers are considered in determinants and inverses of skew circulant and skew left [21]. circulant matrices with Fibonacci and Lucas numbers. Jiang The purpose of this paper is to obtain the explicit et al. [24, 25]consideredtheskewcirculantandskew determinants, explicit inverses, norm, and spread of skew left circulant matrices with the 𝑘-Fibonacci numbers and circulant type matrices involving any continuous Fibonacci the 𝑘-Lucas numbers and discussed the invertibility of the numbers. And we generalize the result [23]. In passing, the these matrices and presented their determinant and the norm and spread of skew circulant type matrices have not inverse matrix by constructing the transformation matrices, been research. It is hoped that this paper will help in changing respectively. Jaiswal evaluated some determinants of circulant this. whose elements are the generalized Fibonacci numbers [26]. In the following, let 𝑟 be a nonnegative integer. We adopt 0 Lind presented the determinants of circulant and skew the following two conventions 0 =1and, for any sequence 𝑛 circulant involving Fibonacci numbers [27]. Dazheng [28] {𝑎𝑛}, ∑𝑘=𝑖 𝑎𝑘 =0in case 𝑖>𝑛. gave the determinant of the Fibonacci-Lucas quasi-cyclic matrices. Definition 1 (see [21]). A skew circulant matrix with the first Recently, there are several papers on the norms of some row (𝑎1,𝑎2,...,𝑎𝑛) is meant to be a square matrix of the form special matrices. Solak [29]establishedthelowerandupper boundsforthespectralnormsofcirculantmatriceswith 𝑎1 𝑎2 ... 𝑎𝑛−1 𝑎𝑛 ˙ classical Fibonacci and Lucas numbers entries. Ipek [30] −𝑎𝑛 𝑎1 𝑎2 ... 𝑎𝑛−1 investigated an improved estimation for spectral norms of . . ( . −𝑎 𝑎 d . ) , these matrices. Shen and Cen [31] gave upper and lower . 𝑛 1 . (3) bounds for the spectral norms of 𝑟-circulant matrices in the . −𝑎3 . dd 𝑎2 form of 𝐴=𝐶𝑟(𝐹0,𝐹1,...,𝐹𝑛−1), 𝐵=𝐶𝑟(𝐿0,𝐿1,...,𝐿𝑛−1), (−𝑎2 −𝑎3 ... −𝑎𝑛 𝑎1 ) andtheyalsoobtainedsomeboundsforthespectralnorms 𝑛×𝑛 𝐴 of Kronecker and Hadamard products of matrix and matrix (𝑎 ,𝑎 ,...,𝑎 ) 𝐵.AkbulakandBozkurt[32] found upper and lower bounds denoted by SCirc 1 2 𝑛 . for the spectral norms of Toeplitz matrices such that 𝑎𝑖𝑗 ≡ Definition 2 (see [21]). A skew left circulant matrix with the 𝐹𝑖−𝑗 and 𝑏𝑖𝑗 ≡𝐿𝑖−𝑗. The convergence in probability and in first row (𝑎1,𝑎2,...,𝑎𝑛) ismeanttobeasquarematrixofthe distribution of the spectral norm of scaled Toeplitz, circulant, form reverse circulant, symmetric circulant, and a class of 𝑘- circulant matrices were discussed in [33]. 𝑎1 𝑎2 𝑎3 ... 𝑎𝑛 Beginning with Mirsky [34], several authors [35–37]have 𝑎 𝑎 ... 𝑎 −𝑎 obtained bounds for the spread of a matrix. 2 3 𝑛 1 ( . ) The Fibonacci sequences are defined by the following 𝑎3 cc c . , (4) recurrence relations [22, 23, 26–32]: . . 𝑎𝑛 −𝑎1 ... −𝑎𝑛−2 𝑎 −𝑎 ... −𝑎 −𝑎 𝐹𝑛+1 =𝐹𝑛 +𝐹𝑛−1 where 𝐹0 =0,𝐹1 =1. (1) ( 𝑛 1 𝑛−2 𝑛−1)𝑛×𝑛 {𝐹 } The 𝑛 is given by the formula denoted by SLCirc(𝑎1,𝑎2,...,𝑎𝑛). 𝛼𝑛 −𝛽𝑛 Lemma 3 {𝐹 } 𝐹 = , (see [30, 31]). Let 𝑛 be Fibonacci numbers; then, 𝑛 𝛼−𝛽 (2) 𝑛−1 2 where 𝛼 and 𝛽 are the roots of the characteristic equation 𝑥 − (𝑖) ∑𝐹𝑖 =𝐹𝑛+1 −1, (5) 𝑥−1=0. 𝑖=0 The Fibonacci sequences were introduced for the first 𝑛−1 2 time by the famous Italian mathematician Leonardo of Pisa (𝑖𝑖) ∑𝐹𝑖 =𝐹𝑛𝐹𝑛−1, (6) (nicknamed Fibonacci). It is well known that the ratio of 𝑖=0 two consecutive classical Fibonacci numbers converges to the 𝑛−1 golden mean, or the golden section, (1+√5)/2,whichappears (𝑖𝑖𝑖) ∑𝑖𝐹𝑖 = (𝑛−1) 𝐹𝑛+1 −𝐹𝑛+2 +2. (7) in modern research in many fields from architecture [38, 39] 𝑖=0 Abstract and Applied Analysis 3
2. Determinant and Inverse of Skew Circulant where Matrix with the Fibonacci Numbers
𝑏1𝑗 =𝐹𝑟+𝑛+2−𝑗, In this section, let 𝐵𝑟,𝑛 = SCirc(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew circulant matrix. Firstly, we give a determinant explicit (12) 𝑏2𝑗 =𝑠𝐹𝑟+𝑛+2−𝑗 −𝐹𝑟+𝑛+3−𝑗, (𝑗=3,4,...,𝑛). formula for the matrix 𝐵𝑟,𝑛.Afterwards,weprovethat𝐵𝑟,𝑛 is an invertible matrix for 𝑛≥2,andthenwefindtheinverseof the matrix 𝐵𝑟,𝑛.Inthefollowing,let So it holds that 𝐹 +𝐹 𝐹 𝑥=− 𝑟 𝑟+𝑛 ,𝑠=𝑟+2 , Γ 𝐵 Π 𝐹𝑟+1 +𝐹𝑟+𝑛+1 𝐹𝑟+1 det det 𝑟,𝑛 det 1 𝑛−2 𝑏=𝐹𝑟+1 +𝐹𝑟+𝑛+1,𝑎=𝐹𝑟 +𝐹𝑟+𝑛, =𝐹𝑟+1 ⋅𝑓𝑛 ⋅𝑏
𝑓𝑛 =𝐹𝑟+1 +𝑠𝐹𝑟+𝑛 =𝐹𝑟+1 [𝐹𝑟+1 +𝑠𝐹𝑟+𝑛 𝑛−2 (8) 𝑛−(𝑘+1) (13) + ∑ (𝑠𝐹𝑟+𝑘+1 −𝐹𝑟+𝑘+2)⋅𝑥 , 𝑛−2 𝑘=1 𝑛−(𝑘+1) + ∑ (𝑠𝐹𝑟+1+𝑘 −𝐹𝑟+2+𝑘)⋅𝑥 ] 𝑛−1 𝑘=1 𝑛−(𝑘+1) 𝑓 = ∑𝐹 ⋅𝑥 . 𝑛 𝑟+𝑘+1 𝑛−2 𝑘=1 ×(𝐹𝑟+1 +𝐹𝑟+𝑛+1) ,
Theorem 4. Let 𝐵𝑟,𝑛 =𝑆𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew (𝑛−1)(𝑛−2)/2 circulant matrix; then, while taking det Γ=det Π1 =(−1) ,wecanget 𝑛−2 det 𝐵𝑟,𝑛 =𝐹𝑟+1 ⋅𝑓𝑛 ⋅𝑏 , (9) 𝐹 𝐵 =𝐹 [ 𝑟+2 ⋅𝐹 +𝐹 where 𝐹𝑟+𝑛 is the (𝑟 + 𝑛)th Fibonacci number. In particular, det 𝑟,𝑛 𝑟+1 𝑟+𝑛 𝑟+1 𝐹𝑟+1 when 𝑟=0,wegettheresultof[23]. 𝑛−2 2 2 𝐹𝑟+2 𝑛−(𝑘+1) Proof. Obviously, det 𝐵𝑟,2 =𝐹 +𝐹 satisfies the equation. + ∑ ( ⋅𝐹 −𝐹 )⋅𝑥 ] 𝑟+1 𝑟+2 𝐹 𝑟+1+𝑘 𝑟+2+𝑘 In case 𝑛>2,let 𝑘=1 𝑟+1 𝑛−2 1 ×(𝐹𝑟+1 +𝐹𝑟+𝑛+1) . 𝑠1 (14) (11−1) ( ) (001−1−1) Γ=( ) , (. ) This completes the proof. (. ccc ) 01cc Theorem 5. Let 𝐵𝑟,𝑛 =𝑆𝐶𝑖𝑟𝑐(𝐹+1,...,𝐹𝑟+𝑛) be a skew 01−1c 0 r circulant matrix; then, 𝐵𝑟,𝑛 is an invertible matrix. Specially, (0 1 −1 −1 ) (10) when 𝑟=0,wegettheresultof[23]. 1 0 0 ⋅⋅⋅ 0 0 𝑛=2 𝐵 =𝐹2 + 𝑛−2 Proof. Taking in Theorem 4, we have det 𝑟,2 𝑟+1 0𝑥 0 ⋅⋅⋅ 0 1 2 𝑛−3 𝐹𝑟+2 =0̸ .Hence𝐵𝑟,2 is invertible. In case 𝑛>2,since𝐹𝑟+𝑛 = 0𝑥 0 ⋅⋅⋅ 1 0 𝑟+𝑛 𝑟+𝑛 ( ) (𝛼 −𝛽 )/(𝛼 − 𝛽),where𝛼+𝛽=1,𝛼𝛽=−1,weobtain Π1 = (. . . . .) . . . d . . 0 𝑥 1 ⋅⋅⋅ 0 𝑘 (0 1 0 ⋅⋅⋅ 0 0) 𝑓(𝜔 𝜂) 𝑛 𝑛×𝑛 𝑘 𝑗−1 be two matrices; then, we have = ∑𝐹𝑟+𝑗(𝜔 𝜂) 𝑗=1 𝐹𝑟+1 𝑓𝑛 𝑏13 ⋅⋅⋅ 𝑏1(𝑛−1) 𝑏1𝑛 𝑛 0𝑓𝑛 𝑏23 ⋅⋅⋅ 𝑏2(𝑛−1) 𝑏2𝑛 1 𝑟+𝑗 𝑟+𝑗 𝑘 𝑗−1 00𝑏 = ∑ (𝛼 −𝛽 )(𝜔 𝜂) ( ) 𝛼−𝛽𝑗=1 Γ𝐵𝑟,𝑛Π1 = ( 00𝑎d ) , (11) . . 1 𝛼𝑟+1 (1 + 𝛼𝑛) 𝛽𝑟+1 (1 + 𝛽𝑛) . . d 𝑏 = [ − ] ( 00 𝑎) 𝑏 𝛼−𝛽 1−𝛼𝜔𝑘𝜂 1−𝛽𝜔𝑘𝜂 4 Abstract and Applied Analysis
𝑟+1 𝑟+1 𝑟+𝑛+1 𝑟+𝑛+1 1 (𝛼 −𝛽 )+(𝛼 −𝛽 ) Then 𝑐𝑖,𝑗 =0,for𝑖<𝑗, 𝑐𝑖𝑖 =𝑔𝑖𝑖𝑔𝑖𝑖 =(𝐹𝑟+1 +𝐹𝑟+𝑛+1)⋅ = [ 1/(𝐹 +𝐹 )=1 𝑖=𝑗 𝛼−𝛽 1−(𝛼+𝛽)𝜔𝑘𝜂+𝛼𝛽𝜔2𝑘𝜂2 𝑟+1 𝑟+𝑛+1 ,for ,and [ 𝑛−2 𝛼𝛽 (𝛼𝑟 −𝛽𝑟 +𝛼𝑟+𝑛 −𝛽𝑟+𝑛)𝜔𝑘𝜂 𝑐𝑖𝑗 = ∑𝑔𝑖𝑘𝑔𝑘𝑗 =𝑔𝑖,𝑖−1𝑔𝑖−1,𝑗 +𝑔𝑖,𝑖𝑔𝑖,𝑗 − ] 𝑘=1 1−(𝛼+𝛽)𝜔𝑘𝜂+𝛼𝛽𝜔2𝑘𝜂2 ] 𝑖−𝑗−1 [−(𝐹 +𝐹 )] =(𝐹 +𝐹 )⋅ 𝑟 𝑟+𝑛 𝑘 𝑟 𝑟+𝑛 𝑖−𝑗 𝐹𝑟+1 +𝐹𝑟+𝑛+1 +(𝐹𝑟 +𝐹𝑟+𝑛)𝜔 𝜂 = (𝐹𝑟+1 +𝐹𝑟+𝑛+1) (21) 1−𝜔𝑘𝜂−𝜔2𝑘𝜂2 𝑖−𝑗 [−(𝐹𝑟 +𝐹𝑟+𝑛)] (𝑘=1,2,...,𝑛−1) , +(𝐹𝑟+1 +𝐹𝑟+𝑛+1)⋅ (𝐹 +𝐹 )𝑖−𝑗+1 (15) 𝑟+1 𝑟+𝑛+1 =0, 𝑙 where 𝜔=exp(2𝜋i/𝑛), 𝜂 = exp(𝜋i/𝑛). If there exists 𝜔 𝜂(𝑙 = 𝑙 𝑖≥𝑗+1 1,2,...,𝑛−1)such that 𝑓(𝜔 𝜂) =,weobtain 0 𝐹𝑟+1 +𝐹𝑟+𝑛+1 + for . 𝑘 𝑙 2𝑙 2 GG−1 =𝐼 𝐼 (𝑛−2)×(𝑛−2) (𝐹𝑟 +𝐹𝑟+𝑛)𝜔 𝜂=0for 1−𝜔𝜂−𝜔 𝜂 =0̸ , and hence it follows Hence, we get 𝑛−2,where 𝑛−2 is an 𝑙 G−1G =𝐼 that 𝜔 𝜂=−(𝐹𝑟+1 +𝐹𝑟+𝑛+1)/(𝐹𝑟 +𝐹𝑟+𝑛) is a real number. Since identity matrix. Similarly, we can verify 𝑛−2.Thus, the proof is completed. (2𝑙 + 1) 𝜋 𝜔𝑙𝜂= ( i) exp 𝑛 Theorem 7. Let 𝐵𝑟,𝑛 =𝑆𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew (16) circulant matrix; then, (2𝑙 + 1) 𝜋 (2𝑙 + 1) 𝜋 = +𝑖 , 1 cos 𝑛 sin 𝑛 −1 𝐵𝑟,𝑛 = ⋅𝑆𝐶𝑖𝑟𝑐(𝑥1,𝑥2,...,𝑥𝑛), (22) 𝑓𝑛 𝑙 it yields that sin((2𝑙 + 1)𝜋/𝑛),sowehave =0 𝜔 𝜂=−1for 0 < (2𝑙 + 1)𝜋/𝑛.Since <2𝜋 𝑥=−1is not the root of the where equation, (−𝑎)𝑛−3 𝑥 =1−(𝐹 −𝑠𝐹 )⋅ 1 𝑟+3 𝑟+2 𝑏𝑛−1 𝐹𝑟+1 +𝐹𝑟+𝑛+1 +(𝐹𝑟 +𝐹𝑟+𝑛)𝑥=0, (𝑛>2) . (17) 𝑛−3 (−𝑎)𝑖−1 𝑘 𝑘 − ∑ (𝐹 −𝑠𝐹 )⋅ , We obtain 𝑓(𝜔 𝜂) =0̸ for any 𝜔 𝜂(𝑘=1,2,...,𝑛−1),while 𝑟+𝑛+2−𝑖 𝑟+𝑛+1−𝑖 𝑖 𝑖=1 𝑏 𝑛 𝑓(𝜂)= ∑𝐹 𝜂𝑗−1 𝑛−2 (−𝑎)𝑖−1 𝑟+𝑗 𝑥 =−𝑠−∑ (𝐹 −𝑠𝐹 )⋅ , 𝑗=1 2 𝑟+𝑛+1−𝑖 𝑟+𝑛−𝑖 𝑏𝑖 (18) 𝑖=1 𝐹𝑟+1 +𝐹𝑟+𝑛+1 +(𝐹𝑟 +𝐹𝑟+𝑛)𝜂 1 = =0.̸ 𝑥 =−(𝐹 −𝑠𝐹 )⋅ , (23) 1−𝜂−𝜂2 3 𝑟+3 𝑟+2 𝑏
It follows from Lemma 3 in [21] that the conclusion holds. 2 (−𝑎)𝑖−1 𝑥 =−∑ (𝐹 −𝑠𝐹 )⋅ , 4 𝑟+1+𝑖 𝑟+𝑖 𝑖 𝑖=1 𝑏 Lemma 6. G =[𝑔 ]𝑛−2 Let the matrix 𝑖,𝑗 𝑖,𝑗=1 be of the form 1 2 (−𝑎)𝑘−5+𝑖 𝑥 =− ∑ (𝐹 −𝑠𝐹 )⋅ , 𝑘 𝑓 𝑟+1+𝑖 𝑟+𝑖 𝑏𝑘−4+𝑖 𝐹 +𝐹 ,𝑖=𝑗, 𝑛 𝑖=1 { 𝑟+1 𝑟+𝑛+1 𝑔 = 𝐹 +𝐹 ,𝑖=𝑗+1, (𝑘=5,6,...,𝑛) . 𝑖𝑗 { 𝑟 𝑟+𝑛 (19) {0, otherwise. In particular, when 𝑟=0,wegettheresultof[23]. G−1 =[𝑔 ]𝑛−2 G Then the inverse 𝑖,𝑗 𝑖,𝑗=1 of is equal to Proof. Let
[−(𝐹 +𝐹 )]𝑖−𝑗 𝑓 { 𝑟 𝑟+𝑛 ,𝑖≥𝑗, 1− 𝑛 𝜋 𝜋 ⋅⋅⋅ 𝜋 𝑖−𝑗+1 𝐹 13 14 1𝑛 𝑔𝑖,𝑗 = {(𝐹 +𝐹 ) (20) 𝑟+1 { 𝑟+1 𝑟+𝑛+1 01𝜋𝜋 ⋅⋅⋅ 𝜋 0, 𝑖 < 𝑗. ( 23 24 2𝑛) { (00 10⋅⋅⋅0) Π2 = ( ) , (24) (00 01⋅⋅⋅0) 𝑟=0 In particular, when ,wegettheresultof[23]...... d . 𝑛−2 . . . . . Proof. Let 𝑐𝑖𝑗 =∑𝑘=1 𝑔𝑖𝑘𝑔𝑘𝑗 . (00 00⋅⋅⋅1) Abstract and Applied Analysis 5 where 1 1 −𝑥4 =𝑈2(𝑛−1) −𝑈2𝑛 = [(𝐹𝑟+2 −𝑠𝐹𝑟+1)⋅ 𝑓𝑛 𝑏 1 𝑓 −𝑎 𝑛 +(𝐹 −𝑠𝐹 )⋅ ] 𝜋1𝑗 = [ (𝑠𝐹𝑟+𝑛+2−𝑗 −𝐹𝑟+𝑛+3−𝑗) 𝑟+3 𝑟+2 2 𝐹𝑟+1 𝑓𝑛 𝑏 1 2 (−𝑎)𝑖−1 −𝐹 ], (𝑗=3,4,...,𝑛), = ∑ (𝐹 −𝑠𝐹 )⋅ , 𝑟+𝑛+2−𝑗 (25) 𝑟+1+𝑖 𝑟+𝑖 𝑖 𝑓𝑛 𝑖=1 𝑏 𝑠𝐹 −𝐹 𝑟+𝑛+2−𝑗 𝑟+𝑛+3−𝑗 −𝑥𝑘 =𝑈2(𝑘−1) −𝑈2𝑘 −𝑈2(𝑘+1) 𝜋2𝑗 =− , (𝑗=3,4,...,𝑛). 𝑓𝑛 1 (−𝑎)𝑘−4 = [(𝐹 −𝑠𝐹 )⋅ 𝑟+2 𝑟+1 𝑘−3 𝑓𝑛 𝑏 Then we have (−𝑎)𝑘−3 +(𝐹 −𝑠𝐹 )⋅ ] 𝑟+3 𝑟+2 𝑏𝑘−2 𝐹𝑟+1 0 0 0 ⋅⋅⋅ 0 0𝑓0 0 ⋅⋅⋅ 0 1 2 (−𝑎)𝑘−5+𝑖 𝑛 = ∑ (𝐹 −𝑠𝐹 )⋅ , 0 0 𝑏 0 ⋅⋅⋅ 0 𝑟+1+𝑖 𝑟+𝑖 𝑘−4+𝑖 ( ) 𝑓𝑛 𝑖=1 𝑏 Γ𝐵𝑟,𝑛Π1Π2 = ( 0 0 𝑎 𝑏 ⋅⋅⋅) 0 . (26) ...... d . . ( 0 0 0 0 ⋅⋅⋅ 𝑏) −𝑥𝑛 =𝑈23 −𝑈24 −𝑈25 1 𝑛−2 (−𝑎)𝑖−1 = ∑ (𝐹 −𝑠𝐹 )⋅ Γ𝐵𝑟,𝑛Π1Π2 =𝐷1 ⊕G,𝐷1 = diag(𝐹𝑟+1,𝑓𝑛) is a diagonal matrix, 𝑟+𝑛+1−𝑖 𝑟+𝑛−𝑖 𝑖 𝑓𝑛 𝑖=1 𝑏 and D1 ⊕ G is the direct sum of D1 and G. If we denote Π= Π Π 𝐵−1 =Π(D−1 ⊕ G−1)Γ 1 2,thenweobtain 𝑟,𝑛 1 . 1 𝑛−3 (−𝑎)𝑖−1 Π − ∑ (𝐹 −𝑠𝐹 )⋅ Since the last row elements of the matrix are 𝑟+𝑛−𝑖 𝑟+𝑛−1−𝑖 𝑖 𝑓𝑛 𝑏 (0, 1, 𝜋23,𝜋24,...,𝜋2(𝑛−1),𝜋2𝑛), then the last row elements of 𝑖=1 Π(D−1 ⊕ G−1) (0, 1/𝑓 ,𝑈 ,𝑈 ,...,𝑈 ) the matrix 1 are 𝑛 23 24 2𝑛 , 1 𝑛−1 (−𝑎)𝑖−1 − ∑ (𝐹 −𝑠𝐹 )⋅ where 𝑟+𝑛−1−𝑖 𝑟+𝑛−2−𝑖 𝑖 𝑓𝑛 𝑖=1 𝑏 1 (−𝑎)𝑛−4 𝑛−2 (−𝑎)𝑖−1 = [(𝐹 −𝑠𝐹 )⋅ 𝑈 = ∑𝜋 ⋅ , 𝑟+2 𝑟+1 𝑛−3 23 2(2+𝑖) 𝑖 𝑓𝑛 𝑏 𝑖=1 𝑏 (−𝑎)𝑛−3 +(𝐹 −𝑠𝐹 )⋅ ] 𝑛+1−𝑘 (−𝑎)𝑖−1 (27) 𝑟+3 𝑟+2 𝑏𝑛−2 𝑈 = ∑ 𝜋 ⋅ , 2𝑘 2(𝑘−1+𝑖) 𝑏𝑖 𝑖=1 1 2 (−𝑎)𝑛−5+𝑖 = ∑ (𝐹 −𝑠𝐹 )⋅ , (𝑘=3,4,...,𝑛) . 𝑟+1+𝑖 𝑟+𝑖 𝑛−4+𝑖 𝑓𝑛 𝑖=1 𝑏 1 𝑥1 = −𝑈23 −𝑈24 −1 𝑓𝑛 Hence it follows from Lemma 6 that letting 𝐵𝑟,𝑛 = 𝑛−2 𝑖−1 SCirc(𝑥1,𝑥2,...,𝑥𝑛) then its last row elements are 1 1 (−𝑎) = − ∑ (𝐹 −𝑠𝐹 )⋅ (−𝑥2,−𝑥3,...,−𝑥𝑛,𝑥1) which are given by the following 𝑟+𝑛+1−𝑖 𝑟+𝑛−𝑖 𝑖 𝑓𝑛 𝑓𝑛 𝑏 equations: 𝑖=1 1 𝑛−3 (−𝑎)𝑖−1 − ∑ (𝐹 −𝑠𝐹 )⋅ 𝑓 𝑟+𝑛+2−𝑖 𝑟+𝑛+1−𝑖 𝑏𝑖 𝑠 𝑠 𝑛−2 (−𝑎)𝑖−1 𝑛 𝑖=1 −𝑥 = +𝑈 = + ∑𝜋 ⋅ 2 23 2(2+𝑖) 𝑖 𝑛−3 𝑓𝑛 𝑓𝑛 𝑏 1 (−𝑎) 𝑖=1 = [1 − (𝐹 −𝑠𝐹 )⋅ 𝑓 𝑟+3 𝑟+2 𝑏𝑛−2 𝑠 1 𝑛−2 (−𝑎)𝑖−1 𝑛 = + ⋅ ∑ (𝐹 −𝑠𝐹 )⋅ , 𝑟+𝑛+1−𝑖 𝑟+𝑛−𝑖 𝑖 𝑛−3 𝑖−1 𝑓𝑛 𝑓𝑛 𝑏 (−𝑎) 𝑖=1 − ∑ (𝐹 −𝑠𝐹 )⋅ ]. 𝑟+𝑛+2−𝑖 𝑟+𝑛+1−𝑖 𝑏𝑖 1 𝐹𝑟+3 −𝑠𝐹𝑟+2 𝑖=1 −𝑥3 =𝑈2𝑛 = ⋅ , 𝑓𝑛 𝑏 (28) 6 Abstract and Applied Analysis
Hence, we obtain 3. Norm and Spread of Skew Circulant Matrix with the Fibonacci Numbers
1 (−𝑎)𝑛−3 Theorem 8. Let 𝐵𝑟,𝑛 =𝑆𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew 𝑥 = [1 − (𝐹 −𝑠𝐹 )⋅ 𝐵 1 𝑓 𝑟+3 𝑟+2 𝑏𝑛−1 circulant matrix; then three kinds of norms of 𝑟,𝑛 are given 𝑛 by 𝑛−3 𝑖−1 (−𝑎) 𝐵 = 𝐵 =𝐹 −𝐹 , − ∑ (𝐹 −𝑠𝐹 )⋅ ], 𝑟,𝑛1 𝑟,𝑛∞ 𝑟+𝑛+2 𝑟+2 (31) 𝑟+𝑛+2−𝑖 𝑟+𝑛+1−𝑖 𝑖 𝑖=1 𝑏 √ 𝐵𝑟,𝑛 = 𝑛(𝐹𝑟+𝑛𝐹𝑟+𝑛+1 −𝐹𝑟𝐹𝑟+1). (32) 𝑠 1 𝑛−2 (−𝑎)𝑖−1 𝐹 𝑥 =− − ∑ (𝐹 −𝑠𝐹 )⋅ , 2 𝑟+𝑛+1−𝑖 𝑟+𝑛−𝑖 𝑖 𝑓𝑛 𝑓𝑛 𝑖=1 𝑏 Proof. By Definition 8 in21 [ ]and(5), we have 𝑛 1 𝐹𝑟+3 −𝑠𝐹𝑟+2 𝑥 =− ⋅ , 𝐵 = 𝐵 = ∑𝐹 =𝐹 −𝐹 . 3 𝑓 𝑏 𝑟,𝑛1 𝑟,𝑛∞ 𝑟+𝑖 𝑟+𝑛+2 𝑟+2 (33) 𝑛 𝑖=1 1 2 (−𝑎)𝑖−1 𝑥 =− ∑ (𝐹 −𝑠𝐹 )⋅ , According to Definition 8 in21 [ ]and(6), we know 4 𝑓 𝑟+1+𝑖 𝑟+𝑖 𝑏𝑖 𝑛 𝑖=1 𝑛 𝑛 2 2 (‖𝐵 ‖ ) = ∑∑𝑎 1 2 (−𝑎)𝑘−5+𝑖 𝑟,𝑛 𝐹 𝑖𝑗 𝑥 =− ∑ (𝐹 −𝑠𝐹 )⋅ , 𝑖=1𝑗=1 𝑘 𝑓 𝑟+1+𝑖 𝑟+𝑖 𝑏𝑘−4+𝑖 𝑛 𝑖=1 𝑛 =𝑛∑𝐹2 . 𝑟+𝑖 . 𝑖=1 (34) 𝑟+𝑛 𝑟 1 2 (−𝑎)𝑛−5+𝑖 2 2 𝑥 =− ∑ (𝐹 −𝑠𝐹 )⋅ , =𝑛(∑𝐹𝑖 − ∑𝐹𝑖 ) 𝑛 𝑟+1+𝑖 𝑟+𝑖 𝑛−4+𝑖 𝑓𝑛 𝑖=1 𝑏 𝑖=0 𝑖=0 1 =𝑛(𝐹 𝐹 −𝐹𝐹 ). 𝐵−1 = ⋅ (𝑥 ,𝑥 ,...,𝑥 ), 𝑟+𝑛 𝑟+𝑛+1 𝑟 𝑟+1 𝑟,𝑛 𝑓 SCirc 1 2 𝑛 𝑛 Thus (29) 𝐵 = √𝑛(𝐹 𝐹 −𝐹𝐹 ). 𝑟,𝑛𝐹 𝑟+𝑛 𝑟+𝑛+1 𝑟 𝑟+1 (35) where
Theorem 9. Let (−𝑎)𝑛−3 𝑥 =1−(𝐹 −𝑠𝐹 )⋅ 𝐵 =𝑆𝐶𝑖𝑟𝑐(𝐹 ,−𝐹 ,...,−𝐹 ,𝐹 ) 1 𝑟+3 𝑟+2 𝑏𝑛−1 𝑟,𝑛 𝑟+1 𝑟+2 𝑟+𝑛−1 𝑟+𝑛 (36) 𝑛−3 (−𝑎)𝑖−1 be an odd-order alternative skew circulant matrix and let 𝑛 be − ∑ (𝐹 −𝑠𝐹 )⋅ , 𝑟+𝑛+2−𝑖 𝑟+𝑛+1−𝑖 𝑖 odd. Then 𝑖=1 𝑏 𝑛 𝑛−2 𝑖−1 𝐵 = ∑𝐹 =𝐹 −𝐹 . (−𝑎) 𝑟,𝑛2 𝑟+𝑖 𝑟+𝑛+2 𝑟+2 (37) 𝑥 =−𝑠−∑ (𝐹 −𝑠𝐹 )⋅ , 𝑖=1 2 𝑟+𝑛+1−𝑖 𝑟+𝑛−𝑖 𝑖 𝑖=1 𝑏 Proof. By Lemma 3 in [21], we have 1 (30) 𝑥3 =−(𝐹𝑟+3 −𝑠𝐹𝑟+2)⋅ , 𝑛 𝑏 𝑖−1 𝑗 𝑖−1 𝜆𝑗 (𝐵𝑟,𝑛)=∑(−1) 𝐹𝑟+𝑖(𝜔 𝜂) . (38) 2 (−𝑎)𝑖−1 𝑖=1 𝑥 =−∑ (𝐹 −𝑠𝐹 )⋅ , 4 𝑟+1+𝑖 𝑟+𝑖 𝑏𝑖 𝑖=1 So 2 𝑘−5+𝑖 𝑛 (−𝑎) 𝑖−1 𝑗 𝑖−1 𝑥 =−∑ (𝐹 −𝑠𝐹 )⋅ , 𝜆 (𝐵 ) ≤ ∑ (−1) 𝐹 ⋅ (𝜔 𝜂) 𝑘 𝑟+1+𝑖 𝑟+𝑖 𝑘−4+𝑖 𝑗 𝑟,𝑛 𝑟+𝑖 𝑖=1 𝑏 𝑖=1 (39) (𝑘=5,6,...,𝑛) . 𝑛 = ∑𝐹𝑟+𝑖, 𝑖=1
This completes the proof. for all 𝑗=0,1,...,𝑛−1. Abstract and Applied Analysis 7
𝑛 Since 𝑛 is odd, ∑𝑖=1 𝐹𝑟+𝑖 is an eigenvalue of 𝐵𝑟,𝑛;thatis furthermore, by (5)and(7),
1 ∑𝑎𝑖𝑗 = (𝑛+2) (𝐹𝑟+𝑛+2 −𝐹𝑟+3) 𝐹𝑟+1 −𝐹𝑟+2 ⋅⋅⋅ 𝐹𝑟+𝑛 −1 𝑖 =𝑗̸ −𝐹𝑟+𝑛 𝐹𝑟+1 ⋅⋅⋅ −𝐹𝑟+𝑛−1 ( 1 ) ( . . . ) (−1) −2[(𝑟+𝑛) 𝐹 −𝐹 − (𝑟+1) 𝐹 . . d . 𝑟+𝑛+2 𝑟+𝑛+3 𝑟+3 (46) . . . . 𝐹 −𝐹 ⋅⋅⋅ 𝐹 . 𝑟+2 𝑟+3 𝑟+1 . +𝐹𝑟+4 −𝑟𝐹𝑟+𝑛+2 +𝑟𝐹𝑟+3] ( 1 ) (40) =2𝐹𝑟+𝑛+4 −𝑛𝐹𝑟+𝑛+2 −𝑛𝐹𝑟+3 −2𝐹𝑟+4. 1 −1 By (19) in [21], we have 𝑛 ( 1 ) = ∑𝐹𝑟+𝑖 ⋅ (−1) . 𝑖=1 . 1 . 𝑠 (𝐵 ) ≥ 2𝐹 −𝑛𝐹 −𝑛𝐹 −2𝐹 . . 𝑟,𝑛 𝑛−1 𝑟+𝑛+4 𝑟+𝑛+2 𝑟+3 𝑟+4 (47) ( 1 )
To sum up, we can get
𝑛 4. Determinant and Inverse of max 𝜆𝑗 (𝐵 ) = ∑𝐹𝑟+𝑖. 0≤𝑗≤𝑛−1 𝑟,𝑛 (41) 𝑖=1 Skew Left Circulant Matrix with the Fibonacci Numbers Since all skew circulant matrices are normal, by Lemma 9 𝐵 = (𝐹 ,...,𝐹 ) in [21], (5), and (41), we obtain In this section, let 𝑟,𝑛 SLCirc 𝑟+1 𝑟+𝑛 be a skew left circulant matrix. By using the obtained conclusions in 𝑛 Section 2, we give a determinant explicit formula for the 𝐵 = ∑𝐹 =𝐹 −𝐹 , 𝐵 𝐵 𝑟,𝑛2 𝑟+𝑖 𝑟+𝑛+2 𝑟+2 (42) matrix 𝑟,𝑛.Afterwards,weprovethat 𝑟,𝑛 is an invertible 𝑖=1 matrix for any positive interger 𝑛.Theinverseofthematrix 𝐵𝑟,𝑛 is also presented. which completes the proof. AccordingtoLemma5in[21], Lemma 6 in [21], and Theorems 4, 5,and7, we can obtain the following theorems. Theorem 10. Let 𝐵𝑟,𝑛 =𝑆𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew circulant matrix; then, the bounds for the spread of 𝐵𝑟,𝑛 are Theorem 11. Let 𝐵𝑟,𝑛 =𝑆𝐿𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew left circulant matrix; then,
𝑠(𝐵𝑟,𝑛)⩽√2𝑛 𝑟+𝑛(𝐹 𝐹𝑟+𝑛+1 −𝐹𝑟+1𝐹𝑟+2), 𝐵 (43) det 𝑟,𝑛 1 𝑠(𝐵𝑟,𝑛)≥ 2𝐹𝑟+𝑛+4 −𝑛𝐹𝑟+𝑛+2 −𝑛𝐹𝑟+3 −2𝐹𝑟+4 . 𝑛−1 𝑛(𝑛−1)/2 𝐹𝑟+2 = (−1) 𝐹𝑟+1 [ ⋅𝐹𝑟+𝑛 +𝐹𝑟+1 𝐹𝑟+1 Proof. The trace of 𝐵𝑟,𝑛 is denoted by tr 𝐵𝑟,𝑛 =𝑛𝐹𝑟+1.By(18) in [21]and(32), we know 𝑛−2 𝐹 + ∑ ( 𝑟+2 ⋅𝐹 −𝐹 )𝑥𝑛−(𝑘+1)] 𝐹 𝑟+1+𝑘 r+2+𝑘 𝑘=1 𝑟+1 𝑠 (𝐵𝑟,𝑛) ⩽ √2𝑛 𝑟+𝑛(𝐹 𝐹𝑟+𝑛+1 −𝐹𝑟+1𝐹𝑟+2). (44) 𝑛−2 ×(𝐹𝑟+1 +𝐹𝑟+𝑛+1) , Since (48)
𝑛 𝑛 𝐹 (𝑟 + 𝑛) ∑𝑎𝑖𝑗 = ∑ (𝑛−(𝑘−1)) 𝐹𝑟+𝑘 − ∑ (𝑘−1) 𝐹𝑟+𝑘 where 𝑟+𝑛 is the th Fibonacci number. 𝑖 =𝑗̸ 𝑘=2 𝑘=2 Theorem 12. Let 𝐵𝑟,𝑛 =𝑆𝐿𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew left 𝑛 𝑛 circulant matrix; then, 𝐵𝑟,𝑛 is an invertible matrix. = (𝑛+2) ∑𝐹𝑟+𝑘 −2∑𝑘𝐹𝑟+𝑘 𝑘=2 𝑘=2 (45) Theorem 13. Let 𝐵𝑟,𝑛 =𝑆𝐿𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew left = (𝑛+2) (𝐹𝑟+𝑛+2 −𝐹𝑟+3) circulant matrix; then,
𝑛 𝑛 −2[∑ (𝑟+𝑘) 𝐹 − ∑𝑟𝐹 ], −1 1 𝑟+𝑘 𝑟+𝑘 𝐵𝑟,𝑛 = 𝑆𝐿𝐶𝑖𝑟𝑐 (𝑥1 ,𝑥2 ,...,𝑥𝑛 ) , (49) 𝑘=2 𝑘=2 𝑓𝑛 8 Abstract and Applied Analysis where By (55)and(56), we know (−𝑎)𝑛−3 𝑛 𝑥 =1−(𝐹𝑟+3 −𝑠𝐹𝑟+2)⋅ 1 𝑛−2 max 𝜆𝑖 (𝐵𝑟,𝑛) = ∑𝐹𝑟+𝑖. (57) 𝑏 0≤𝑖≤(𝑛+1)/2 𝑖=1 𝑛−3 (−𝑎)𝑖−1 − ∑ (𝐹 −𝑠𝐹 )⋅ , Since all skew left circulant matrices are symmetrical, by 𝑟+𝑛+2−𝑖 𝑟+𝑛+1−𝑖 𝑖 𝑖=1 𝑏 Lemma 9 in [21], (5), and (57), we obtain 𝐵𝑟,𝑛 =𝐹𝑟+𝑛+2 −𝐹𝑟+2. (58) 𝑥𝑘 =−𝑥𝑛−𝑘+2 2
2 (−𝑎)𝑛−𝑘−3+𝑖 = ∑ (𝑓 −𝑡𝑓 )⋅ , 𝑟+1+𝑖 𝑟+𝑖 𝑏𝑛−𝑘−2+𝑖 Theorem 16. 𝐵 =𝑆𝐿𝐶𝑖𝑟𝑐(𝐹 ,...,𝐹 ) 𝑖=1 (50) Let 𝑟,𝑛 𝑟+1 𝑟+𝑛 be skew left circulant matrix, if 𝑛 is odd, then (𝑘=2,3,...,𝑛−2) , 2𝐹𝑟+𝑛 ≤𝑠(𝐵 ) 1 𝑟,𝑛 𝑥 =−𝑥 =(𝐹 −𝑠𝐹 )⋅ , 𝑛−1 3 𝑟+3 𝑟+2 𝑏 2 √ 2 ⩽ 2𝑛 𝑟+𝑛(𝐹 𝐹𝑟+𝑛+1 −𝐹𝑟+1𝐹𝑟)− (𝐹𝑟+𝑛−1 +𝐹𝑟−1) ; 𝑥 =−𝑥 𝑛 𝑛 2 (59) 𝑛−2 (−𝑎)𝑖−1 𝑛 =𝑠+∑ (𝐹 −𝑠𝐹 )⋅ . if is even, then 𝑟+𝑛+1−𝑖 𝑟+𝑛−𝑖 𝑏𝑖 𝑖=1 √ 2𝐹𝑟+𝑛 ≤𝑠(𝐵𝑟,𝑛) ⩽ 2𝑛 𝑟+𝑛(𝐹 𝐹𝑟+𝑛+1 −𝐹𝑟+1𝐹𝑟). (60)
5. Norm and Spread of Skew Left Circulant Proof. Since 𝐵𝑟,𝑛 is a symmetric matrix, by (20) in [21], Matrix with the Fibonacci Numbers 𝑠(𝐵 )≥2max 𝑎𝑖𝑗 =2𝐹𝑟+𝑛. 𝑟,𝑛 𝑖 =𝑗̸ (61) Theorem 14. Let 𝐵𝑟,𝑛 =𝑆𝐿𝐶𝑖𝑟𝑐(𝐹𝑟+1,...,𝐹𝑟+𝑛) be a skew left circulant matrix. Then three kinds of norms of 𝐵 are given by 𝑟,𝑛 If 𝑛 is odd, the trace of 𝐵𝑟,𝑛 is 𝐵𝑟,𝑛 = 𝐵𝑟,𝑛 =𝐹𝑟+𝑛+2 −𝐹𝑟+2, 1 ∞ tr (𝐵𝑟,𝑛) (51) 𝐵 = √𝑛(𝐹 𝐹 −𝐹𝐹 ). =𝐹 −𝐹 +𝐹 −⋅⋅⋅+𝐹 𝑟,𝑛𝐹 𝑟+𝑛 𝑟+𝑛+1 𝑟 𝑟+1 𝑟+1 𝑟+2 𝑟+3 𝑟+𝑛
Proof. Using the method in Theorem 8 similarly, the conclu- =𝐹𝑟+1 +𝐹𝑟+1 +𝐹𝑟+3 +⋅⋅⋅+𝐹𝑟+𝑛−2 sion is obtained. (62) =2𝐹𝑟+1 +𝐹𝑟+1 +𝐹𝑟+2 +⋅⋅⋅+𝐹𝑟+𝑛−3 Theorem 15. Let 𝑛−3 =2𝐹𝑟+1 + ∑𝐹𝑟+𝑖; 𝐵𝑟,𝑛 =𝑆𝐿𝐶𝑖𝑟𝑐(𝐹𝑟+1,−𝐹𝑟+2,...,−𝐹𝑟+𝑛−1,𝐹𝑟+𝑛) (52) 𝑖=1 be an odd-order alternative skew left circulant matrix; then, by (5), we know 𝑛 𝐵 = ∑𝐹 =𝐹 −𝐹 . (𝐵 )=𝐹 +𝐹 . 𝑟,𝑛2 𝑟+𝑖 𝑟+𝑛+2 𝑟+2 (53) tr 𝑟,𝑛 𝑟+𝑛−1 𝑟−1 (63) 𝑖=1 By (18) in [21], (51), and (63), we obtain Proof. According to Lemma 4 in [21], 𝑠(𝐵 ) 𝑛 𝑟,𝑛 𝑖−1 (𝑗−(1/2))(𝑘−1) 𝜆𝑗 (𝐵𝑟,𝑛)=±∑(−1) 𝐹𝑟+𝑖𝜔 , (54) 2 (64) 𝑖=1 ⩽ √2𝑛 (𝐹 𝐹 −𝐹𝐹 )− (𝐹 +𝐹 )2. 𝑟+𝑛 𝑟+𝑛+1 𝑟 𝑟+1 𝑛 𝑟+𝑛−1 𝑟−1 for 𝑗=1,2,...,(𝑛−1)/2,and 𝑛 If 𝑛 is even, the trace of 𝐵𝑟,𝑛 is 𝜆 (𝐵 )=∑𝐹 . (𝑛+1)/2 𝑟,𝑛 𝑟+𝑖 (55) 𝑖=1 tr (𝐵𝑟,𝑛)=𝐹𝑟+1 −𝐹𝑟+1 +𝐹𝑟+3 (65) So −𝐹𝑟+3 ⋅⋅⋅+ 𝐹𝑟+𝑛−1 −𝐹𝑟+𝑛−1 =0. 𝑛 𝜆 (𝐵 ) ≤ ∑ (−1)𝑖−1𝐹 (−1)𝑖−1 𝑗 𝑟,𝑛 𝑟+𝑖 By (18) in [21], (51), and (65), we can get 𝑖=1 (56) 𝑠 (𝐵 ) ⩽ √2𝑛 (𝐹 𝐹 −𝐹𝐹 ). 𝑛 𝑛+1 𝑟,𝑛 𝑟+𝑛 𝑟+𝑛+1 𝑟 𝑟+1 (66) = ∑𝐹𝑟+𝑖, (𝑗 = 1, 2, ⋅ ⋅ ⋅, ). 𝑖=1 2 So the result follows. Abstract and Applied Analysis 9
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