Explicit Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with the Pell-Lucas Numbers Abstract

Journal of Advances in Mathematics and Computer Science

26(2): 1-16, 2018; Article no.JAMCS.38768 ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-0851)

Explicit Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with the Pell-Lucas Numbers

∗ Jinjiang Yao1 and Jixiu Sun1,2

1School of Mathematics and Statistics, Linyi University, Linyi 276005, P. R. China. 2School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, P.R. China.

Authors’ contributions

This work was carried out in collaboration between all authors. Author JY designed the study, performed the statistical analysis and wrote the ﬁrst draft of the manuscript. Author JS managed the analyses of the study. All authors read and approved the ﬁnal manuscript.

Article Information DOI: 10.9734/JAMCS/2018/38768 Editor(s): (1) Dariusz Jacek Jakobczak, Assistant Professor, Chair of Computer Science and Management in this Department, Technical University of Koszalin, Poland. Reviewers: (1) Omur Deveci, Kafkas University, Turkey. (2) Grienggrai Rajchakit, Maejo University, Thailand. (3) Ahmet Oteles, Dicle University, Turkey. (4) Mustafa Bahsi, Aksaray University, Turkey. Complete Peer review History: http://www.sciencedomain.org/review-history/22872

Received: 12th November 2017 Accepted: 17th January 2018 Original Research Article Published: 26th January 2018

Abstract

In this paper, we consider the skew circulant and skew left circulant matrices with the Pell-Lucas numbers. We discuss the invertibility of the skew circulant and skew left circulant matrices and present the determinant and the inverse matrix of them by constructing the transformation matrices.

Keywords: Skew circulant matrix; skew left circulant matrix; determinant; inverse; Pell-Lucas number.

2010 Mathematics Subject Classiﬁcation: 15A09, 15A15, 15B05, 11B39.

*Corresponding author: E-mail: [email protected]; Yao and Sun; JAMCS, 26(2): 1-16, 2018; Article no.JAMCS.38768

1 Introduction

Skew circulant and circulant matrices have important applications in various disciplines [1], [2], [3], such as image processing, communications, signal processing, encoding, preconditioner, and solving least squares problems. In [4], [5], [6],[7], the skew-circulant matrix as pre-conditioners for LMF- based ODE codes, Hermitian and skew-Hermitian Toeplitz systems were considered. Lyness [8] constructed an s-dimensional lattice rules by a skew-circulant matrix. In [9], the authors discussed the spectral decompositions of skew circulant and skew left circulant matrices.

Recently, there are several papers on the Pell sequences and circulant matrices. M. Akbulak et al. [10] gave sum formulas of Pell and Jacobsthal sequences by matrix method. A. Oteles¨ et al. investigated permanents of an n × n (0,1,2)-matrix by contraction method and showed that the permanent of the matrix is equal to the generalized k-Pell numbers in [11]. In [12], [13], [14], the authors deﬁned the recurrence sequences by using the circulant matrices which are obtained from the characteristic polynomial.

The Pell-Lucas sequences are deﬁned by the following recurrence relations, respectively [15, 16, 17, 18]: Qn+1 = 2Qn + Qn−1, where Q0 = 2,Q1 = 2, for n ≥ 0, the ﬁrst few values of the sequences are given by the following table: n 0 1 2 3 4 5 6 7 8 9 Qn 2 2 6 14 34 82 198 478 1154 2786

n n The {Qn} is given by the formula Qn = α + β , where α and β are the roots of the characteristic equation x2 − 2x − 1 = 0.

The purpose of this paper is to obtain the determinants and inverses of skew circulant and skew left circulant matrices by using some properties of Pell and Pell-Lucas numbers . In the following, 0 let r be∑ a nonegative integer. We adopt the following two conventions 0 = 1, and for any sequence { } n an , k=i ak = 0 if i > n.

Deﬁnition 1.1 ([9]). A skew circulant matrix with the ﬁrst row (a1, a2, . . . , an) is a square matrix of the form   a1 a2 . . . an−1 an  −   an a1 a2 an−1     . .. .   . −an a1 . .     .. ..  −a3 . . a2 −a −a ... −a a 2 3 n 1 n×n denoted by SCirc(a1, a2, ..., an).

Deﬁnition 1.2 ([9]). A skew left circulant matrix with the ﬁrst row (a1, a2, . . . , an) is a square matrix of the form   a1 a2 . . . an−1 an  −   a2 an−1 an a1     ......   . . . . .    an−1 an −a1 −an−2 a −a ... −a − −a − n 1 n 2 n 1 n×n denoted by SLCirc(a1, a2, ..., an).

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Lemma 1.1 ([1], [9]). Let A := SCirc(a1, a2, ..., an). Then we have ∑n k j−1 (i) A is invertible if and only if f(ω η) ≠ 0 (k = 0, 1, 2, ..., n − 1), where f(x) = aj x , j=1 2πi πi ω = exp( n ) and η = exp( n ); (ii) if A is invertible, then the inverse of A is a skew circulant matrix.

Lemma 1.2 ([19]). Let A := SLCirc(a1, a2, ..., an) and n be odd. Then

∑n (j− 1 )(k−1) n − 1 λ = ±| a ω 2 |, (j = 1, 2,..., ), j k 2 k=1 ∑n k−1 λ n+1 = |ak(−1) |, 2 k=1

n−1 n+1 where λj (j = 1, 2,..., 2 , 2 ) are the eigenvalues of A.

Lemma 1.3. For the orthogonal skew left circulant matrix   1 0 ······ 0    . .   0 . .. −1     . . .  Θ :=  . . . −  ,  . . . 1 0   . . . .   . 0 .. .. .  0 −1 0 ··· 0 n×n we have

SCirc(a1, a2, . . . , an) = ΘSLCirc(a1, a2, . . . , an).

−1 Lemma 1.4. If [SCirc(a1, a2, ..., an)] = SCirc(b1, b2, ..., bn), then

−1 [SLCirc(a1, a2, ..., an)] = SLCirc(b1, −bn, ..., −b2).

′ Proof. Let B := SCirc(a1, a2, ..., an) and A := SLCirc(a1, a2, ..., an). By Lemma 1.3, we have B = ′ ′−1 −1 −1 ΘA . Thus, we obtain A = B Θ = SLCirc(b1, −bn, ..., −b2), where B = SCirc(b1, b2, ..., bn).

Lemma 1.5 ([20]). Let {Qn} be Pell-Lucas numbers. Then

n−1 ∑ 1 (i) Q = (Q + Q − ), (1.1) i 2 n n 1 i=0 n∑−1 2 1 (ii) Q = Q Q − + 2, (1.2) i 2 n n 1 i=0 n−1 ∑ 1 (iii) iQ = [(n − 2)Q + (n − 1)Q − + 2]. (1.3) i 2 n n 1 i=0

Proof. Since Q0 = 2, Q1 = 2 and by [20], we have (i) and (ii).

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(iii) By Equation (1.1),

n∑−1 n∑−1 n∑−1 n∑−1 n∑−1 n−∑k−1 iQi = Qi = ( Qi − Qi) i=1 k=1 i=n−k k=1 i=0 i=0 n−1 1 ∑ = (Q + Q − − Q − − Q − − ) 2 n n 1 n k n k 1 k=1 1 = [(n − 2)Q + (n − 1)Q − + 2]. 2 n n 1 The proof is completed.

Deﬁnition 1.3 ([21]). Let A = (aij ) be an n × n matrix. The Euclidean (or Frobenius) norm, the spectral norm, the maximum column sum matrix norm and the maximum row sum matrix norm of the matrix A are, respectively:

( ) 1 ∑n 2 2 ∥A∥F = |aij | , (1.4) i,j=1 ( ) 1 ∗ 2 ∥A∥2 = max λi(A A) , (1.5) 1≤i≤n ∑n ∥A∥1 = max |aij |, (1.6) 1≤j≤n i=1 ∑n ∥A∥∞ = max |aij |, (1.7) 1≤i≤n j=1 where A∗ denotes the conjugate transpose of A. Lemma 1.6 ([22]). If A is an n × n real symmetric or normal matrix, then we have

∥A∥2 = max |λi|, (1.8) 1≤i≤n where λi (i = 1, 2, . . . , n) are the eigenvalues of A.

Deﬁnition 1.4 ([23]). Let A = (aij ) be an n × n matrix with eigenvalues λi (i = 1, 2, ...n). The spread of A is deﬁned as

s(A) = max | λi − λj | . (1.9) i,j In [23], authors have obtained an upper bound for the spread of a matrix, which states that √ 2 s(A) ≤ 2 ∥ A ∥2 − | trA |2, (1.10) F n where ∥ A ∥F denotes the Frobenius norm of A and trA is trace of A.

Lemma 1.7 ([24]). Let A = (aij ) be an n × n matrix. Then (i) if A is real and normal, then 1 ∑ s(A) ≥ | a |; (1.11) n − 1 ij i≠ j (ii) if A is Hermitian, then

s(A) ≥ 2 max |aij |. (1.12) i≠ j

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2 Determinant and Inverse of Skew Circulant Matrix with the Pell-Lucas Numbers

In this section, let Dr,n := SCirc(Qr+1,Qr+2,...,Qr+n). Firstly, we give a determinant explicit formula for the matrix Dr,n. Afterwards, we prove that Dr,n is an invertible matrix for any positive integer n, and then we ﬁnd the inverse of the matrix Dr,n.

Theorem 2.1. Let Dr,n := SCirc(Qr+1,Qr+2,...,Qr+n). Then we have [ ] n∑−2 n−(k+1) n−2 det Dr,n = Qr+1 · Qr+1 + θQr+n + (θQr+k+1 − Qr+k+2)·y · (Qr+1 + Qr+n+1) , k=1 (2.1) Qr+2 Qr +Qr+n where Qr+n is the (r + n)th Pell-Lucas number, θ = and y = − . Qr+1 Qr+1+Qr+n+1

2 2 Proof. Obviously, det Dr,2 = Qr+1 + Qr+2 satisﬁes the equation (1.1). When n > 2, we introduce two n × n matrices as follows:   1 0 ··············· 0    . .     θ . .. 1  1 0 0 ······ 0    . .   . .   . .. −   n−2 . ..   1 . 1 2   0 (y) . 1   .   . .   . ..   . n−3 . ..   0 . . 1 −2 −1   .(y) . . 1 0  Σ =   and Ω1 =   ,  ......   . . . . .   ...... 0   . . 0 .. .. .       . . .   . .   ......   . .. .   . . . 1 . . . .  . y 1 . .  . . . .  0 1 0 ······ 0  . 0 1 −2 .. .. .  0 1 −2 −1 0 ······ 0 Q + Q where y = − r r+n . Qr+1 + Qr+n+1

And we have

 ′  Qr+1 qn Qr+n−1 ······ Qr+3 Qr+2  − ······ − −   0 qn θQr+n−1 Qr+n θQr+3 Qr+4 θQr+2 Qr+3     .   . 0 Q + Q 0 ··· ··· 0   r+1 r+n+1   . . . . .   . . . . .  ΣDr,nΩ1 =  . . Qr + Qr+n . . .  ,    ......   . . 0 . . . .     ......  . . . . . Qr+1 + Qr+n+1 0 0 0 0 ··· 0 Qr + Qr+n Qr+1 + Qr+n+1 where

n∑−2 n∑−1 Qr+2 n−(k+1) ′ n−(k+1) θ = , qn = Qr+1 + θQr+n + (θQr+1+k − Qr+2+k)·y , qn = Qr+k+1·y . Qr+1 k=1 k=1

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Hence, we obtain − det Σ det D det Ω = Q q · (Q + Q )n 2 r,n 1 [ r+1 n r+1 r+n+1 ] n∑−2 n−(k+1) n−2 = Qr+1 · Qr+1 + θQr+n + (θQr+k+1 − Qr+k+2)·y · (Qr+1 + Qr+n+1) . k=1 While (n−1)(n−2) det Σ = det Ω1 = (−1) 2 , we have [ ] n∑−2 n−(k+1) n−2 det Dr,n = Qr+1 · Qr+1 + θQr+n + (θQr+k+1 − Qr+k+2)·y · (Qr+1 + Qr+n+1) . k=1

Theorem 2.2. Let Dr,n := SCirc(Qr+1,Qr+2,...,Qr+n). Then Dr,n is invertible for any positive integer n . r+n r+n · − πi Proof. Since Qr+n = α + β , where α + β = 2, α β = 1 and η = exp( n ), we have ∑n ∑n k k j−1 r+j r+j k j−1 f(ω η) = Qr+j (ω η) = (α + β )(ω η) j=1 j=1 αr(1 + αn) βr(1 + βn) = + (because 1 − αωkη ≠ 0, 1 − βωkη ≠ 0) 1 − αωkη 1 − βωkη (αr+1 + βr+1) + (αr+n+1 + βr+n+1) − αβ(αr+n + βr+n)ωkη − αβ(αr + βr)ωkη = 1 − (α + β)ωkη + αβω2kη2 Q + Q + (Q + Q )ωkη = r+1 r+n+1 r r+n , (k = 1, 2, ..., n − 1). 1 − 2ωkη − ω2kη2 l l If there exists ω η(l = 1, 2, . . . , n − 1) such that f(ω η) = 0, then we obtain Qr+1 + Qr+n+1 + (Qr + l l 2l 2 l Qr+1+Qr+n+1 Qr+n)ω η = 0 for 1 − 2ω η − ω η ≠ 0, thus, ω η = − is a real number, while Qr +Qr+n (2l + 1)πi (2l + 1)π (2l + 1)π ωlη = exp = cos + i sin . n n n

(2l+1)π l − (2l+1)π − Hence, sin n = 0, and we have ω η = 1 for 0 < n < 2π. But x = 1 is not the root of the equation Qr+1 + Qr+n+1 + (Qr + Qr+n)x = 0 for any positive integer n. Hence, we obtain f(ωkη) ≠ 0 for any ωkη(k = 1, 2, . . . , n − 1), while ∑n − Q + Q + (Q + Q )η f(η) = Q ηj 1 = r+1 r+n+1 r r+n ≠ 0. j 1 − 2η − η2 j=1 Thus, by Lemma 1.1, the proof is completed. H n−2 Lemma 2.3. Let the matrix = [hij ]i,j=1 be of the form   Qr+1 + Qr+n+1, i = j, hij =  Qr + Qr+n, i = j + 1, 0, otherwise. H−1 ′ n−2 Then = [hij ]i,j=1 is { i−j (−Qr −Q ) ′ r+n ≥ (Q +Q )i−j+1 , i j, hij = r+1 r+n+1 0, i < j.

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n−2 ∑ ′ Proof. Let rij = hikhkj . Obviously, rij = 0 for i < j. For i = j, we obtain k=1

′ 1 rii = hiihii = (Qr+1 + Qr+n+1) · = 1. Qr+1 + Qr+n+1

For i ≥ j + 1, we obtain

n∑−2 ′ ′ ′ rij = hikhkj = hi(i−1)h(i−1)j + hiihij k=1 − − i−j−1 − − i−j · ( Qr Qr+n) · ( Qr Qr+n) = (Qr + Qr+n) i−j + (Qr+1 + Qr+n+1) i−j+1 = 0. (Qr+1 + Qr+n+1) (Qr+1 + Qr+n+1)

−1 Hence, we verify HH = In−2, where In−2 is (n − 2) × (n − 2) identity matrix. Similarly, we can −1 verify H H = In−2. Thus, the proof is completed.

Theorem 2.4. Let Dr,n := SCirc(Qr+1,Qr+2,...,Qr+n). Then we have

′ ′ −1 1 ′ Dr,n = SCirc(y1, y2, . . . , y n), qn

where

n∑−3 − − ′ (−d)i 1 (−d)n 3 y = 1 − (Q − − θQ − )· − 2(Q − θQ ) , 1 r+n+2 i r+n+1 i ci r+3 r+2 cn−2 i=1 n∑−2 − ′ (−d)i 1 y = −θ − (Q − − Q − ) , 2 r+n+1 i r+n i ci i=1 ′ 1 y = −(Q − θQ ) · , 3 r+3 r+2 c ∑2 − ′ (−d)i 1 y = − (Q − θQ ) , 4 r+1+i r+i ci i=1 ···

∑2 − ′ (−d)k 5+i y = − (Q − θQ )· , (k = 5, 6, . . . , n), k r+1+i r+i (c)k−4+i i=1

Qr+2 θ = , c = Qr+n+1 + Qr+1, d = Qr+n + Qr. Qr+1

Proof. Let  ′  qn 1 − µ1,3 µ1,4 ··· µ1,n  Qr+1   ···   0 1 µ2,3 µ2,4 µ2,n     . .   . .. 1 0 ··· 0    Ω2 =  . . . .  ,  ......   . 1 .   . . .   . .. .. 0  0 ········· 0 1

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where ′ 1 qn µ1,3 = [ (θQr+n−1 − Qr+n) − Qr+n−1], Qr+1 qn ′ 1 qn µ1,k = [ (θQr+n−(k−2) − Qr+n−(k−3) − Qr+n−(k−2)], (k = 4, 5, . . . , n), Qr+1 qn

Qr+n − θQr+n−1 µ2,3 = , qn

Qr+n+3−k − θQr+n+2−k µ2,k = , (k = 4, 5, . . . , n), qn n∑−1 Qr+2 ′ n−(k+1) θ = , qn = Qr+k+1·y , Qr+1 k=1 n∑−2 n−(k+1) qn = Qr+1 + θQr+n + (θQr+k+1 − Qr+k+2)·y . k=1 Then we have   Qr+1 0 ············ 0    . .   0 q .. .   n   . .   . .. .   . 0 c . .     . . .. .  M ⊕ H ΣDr,nΩ1Ω2 =  . . d c . .  = ,    ......   ......   . . 0 .   ......   ...... 0  0 0 0 ··· 0 d c where c = Qr+n+1 + Qr+1, d = Qr+n + Qr, M = diag(Qr+1, qn) is a diagonal matrix, and M ⊕ H is the direct sum of M and H.

If we denote Ω = Ω1Ω2, then we obtain

−1 −1 −1 Dr,n = Ω(M ⊕ H )Σ.

Since the last row elements of the matrix Ω are (0, 1, µ2,3, µ2,4, . . . , µ2,n), the last row elements of −1 −1 1 the matrix Ω(M ⊕ H ) are (0, , ν2,3, ··· , ν2,n), where qn

n−2 ∑ (−d)i−1 ν = µ · , 2,3 2,2+i ci i=1 n−(k−1) ∑ (−d)i−1 ν = µ − · , (k = 4, 5, . . . , n). 2,k 2,k 1+i ci i=1

−1 −1 Hence, the last row elements of the matrix Ω(M ⊕ H )Σ are (ω1, ω2, . . . , ωn), where 1 ω1 = θ + ν23, ω2 = ν2n, ω3 = ν2n−1 − 2ν2n, qn

ωk = ν2,n−k+2 − 2ν2,n−k+3 − ν2,n−k+4, (k = 4, 5, . . . , n − 1), 1 ωn = − 2ν23 − ν24. qn

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−1 By Lemma 1.5, if let Dr,n := SCirc(y1, y2, . . . , yn), then its last row elements (−y2, −y3,..., −yn, y1), and are given by the following equations: 1 y1 = ωn = − 2ν2,3 − ν2,4 qn n−3 1 ∑ (−d)i−1 (−d)n−3 − − − − · − − = [1 (Qr+n+2 i θQr+n+1 i) i 2(Qr+3 θQr+2) n−2 ], qn c c i=1 n−2 1 1 1 ∑ (−d)i−1 − − − − y2 = ω1 = θ + ν2,3 = θ + (Qr+n+1 i Qr+n i) i qn qn qn c i=1 1 1 −y3 = ω2 = ν2,n = (Qr+3 − θQr+2) · , qn c 1 −d 1 −y = ω = ν − − 2ν = µ − + µ − 2µ 4 3 2,n 1 2,n 2,n 1 c 2,n c2 2,n c ∑2 − i−1 1 · − ( d) = (Qr+1+i θQr+i) i , qn c i=1

−yk = ν2,n−k+3 − 2ν2,n−k+4 − ν2,n−k+5 k−2 k−3 1 ∑ (−d)i−1 2 ∑ (−d)i−1 − − − · − − − − − · = (Qr+k+1 i θQr+k i) i (Qr+k i θQr+k 1 i) i qn c qn c i=1 i=1 k−4 1 ∑ (−d)i−1 − − − − − − · (Qr+k 1 i θQr+k 2 i) i qn c i=1 ∑2 − k−5+i 1 − · ( d) = (Qr+1+i θQr+i) k−4+i , (k = 5, 6, . . . , n). qn c i=1

Hence, we obtain

′ ′ ′ −1 1 Dr,n = SCirc(y1, y2, . . . , yn), qn where n∑−3 − − ′ (−d)i 1 (−d)n 3 y = 1 − (Q − − θQ − )· − 2(Q − θQ ) , 1 r+n+2 i r+n+1 i ci r+3 r+2 cn−2 i=1 n∑−2 − ′ (−d)i 1 y = −θ − (Q − − Q − ) , 2 r+n+1 i r+n i ci i=1 ′ 1 y = −(Q − θQ ) · , 3 r+3 r+2 c ∑2 − ′ (−d)i 1 y = − (Q − θQ ) , 4 r+1+i r+i ci i=1 ···

∑2 − ′ (−d)k 5+i y = − (Q − θQ )· , (k = 5, 6, . . . , n). k r+1+i r+i ck−4+i i=1

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3 Norm and Spread of Skew Circulant Matrix with the Pell Lucas Numbers

Theorem 3.1. Let Dr,n := SCirc(Qr+1,Qr+2,...,Qr+n). Then three kinds norms of Dr,n are given by 1 ∥Dr,n∥1 = ∥Dr,n∥∞ = (Qr+n+1 + Qr+n − Qr+1 − Qr), (3.1) √ 2 n ∥D ∥ = (Q Q − Q Q ). (3.2) r,n F 2 r+n+1 r+n r+1 r

Proof. On the basis of the deﬁnitions of norms, we have

n ∑ 1 ∥D ∥ = ∥D ∥∞ = Q = (Q + Q − Q − Q ). r,n 1 r,n r+i 2 r+n+1 r+n r+1 r i=1

n∑−1 2 1 By (1.2), we have Qi = 2 (Qn+1Qn) + 2. Hence i=0

∑n ∑n ∑n 2 2 2 ∥ Dr,n ∥F = | aij | = n Qr+i i=1 j=1 i=1 r∑+n ∑r 2 2 = n( Qi − Qi ) i=0 i=0 n = (Q Q − Q Q ). 2 r+n+1 r+n r+1 r Thus √ n ∥D ∥ = (Q Q − Q Q ). r,n F 2 r+n+1 r+n r+1 r

′ Theorem 3.2. Let Dr,n = SCirc(Qr+1, −Qr+2,..., −Qr+n−1,Qr+n) be an odd-order alternative skew-circulant matrix, and 1 = n(mod 2). Then

∑n ′ 1 ∥D ∥ = Q = (Q + Q − Q − Q ). r,n 2 r+i 2 r+n+1 r+n r+1 r i=1

′ Proof. We employ [23] and [22] to calculate the spectral norm of Dr,n as follows, for all j = 0, 1, . . . , n − 1:

∑n ′ | | − i−1 j i−1 λj (Dr,n) = ( 1) Qr+i(ω α) i=1 ∑n ∑n i−1 j i−1 ≤ |(−1) Qr+i| · |(ω α) | = Qr+i. i=1 i=1

′ ′ Since all skew-circulant matrices are normal, we deduce that ∥Dr,n∥2 = max λj (Dr,n) . 0≤j≤n−1

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∑n ′ If n is odd, then Qr+i is an eigenvalue of Dr,n as follows i=1       Qr+1 −Qr+2 Qr+3 ··· Qr+n 1 1    .   −1   −1   −Q Q −Q . −Q −       r+n r+1 r+2 r+n 1   1  ∑n  1   .       − .   −1  = Qr+i ·  −1  .  Qr+n−1 Qr+n Qr+1 . Qr+n−2        i=1    . . . . .   .   .   . . . . .  . . Qr+2 −Qr+3 Qr+4 ··· Qr+1 1 1

∑n ′ Noticing that Lemma 1.2, we claim that Qr+i is the maximum of |λj (Dr,n)|, and we obtain i=1 ∑n ′ 1 ∥D ∥ = Q = (Q + Q − Q − Q ). r,n 2 r+i 2 r+n+1 r+n r+1 r i=1 This completes the proof.

Theorem 3.3. Let Dr,n := SCirc(Qr+1,Qr+2, ..., Qr+n). Then the bounds for the spread of Dr,n are √ s(Dr,n) ≤ n(Qr+nQr+n+1 − Qr+1Qr+2) and 1 s(D ) ≥ |(n − 4)Q + (n − 2)Q + (n + 2)Q + nQ |. r,n 2(n − 1) r+n+1 r+n r+2 r+1 Proof. We use the result of Theorem 3.1, n ∥ D ∥2 = (Q Q − Q Q ), r,n F 2 r+n+1 r+n r r+1 and trDr,n = nQr+1. Hence √ s(Dr,n) ≤ n(Qr+nQr+n+1 − Qr+1Qr+2). Since n∑−1 n∑−1 n∑−1 n∑−1 n∑−1 n−∑k−1 iQi = Qi = ( Qi − Qi) i=1 k=1 i=n−k k=1 i=0 i=0 1 = [(n − 2)Q + (n − 1)Q − + 2], 2 n n 1 it yields that ∑ ∑n ∑n aij = (n − (k − 1))Qr+k − (k − 1)Qr+k i≠ j k=2 k=2 ∑n ∑n = (n + 2) Qr+k − 2 kQr+k k=2 k=2 ∑n ∑n = (n + 2r + 2) Qr+k − 2 (r + k)Qr+k k=2 k=2 1 = (n + 2r + 2)(Q + Q − Q − Q ) 2 r+n+1 r+n r+2 r+1

− [(r + n − 1)Qr+n+1 + (r + n)Qr+n − rQr+2 − (r + 1)Qr+1] 1 = [(4 − n)Q + (2 − n)Q − (n + 2)Q − nQ ]. 2 r+n+1 r+n r+2 r+1

11 Yao and Sun; JAMCS, 26(2): 1-16, 2018; Article no.JAMCS.38768

Thus 1 s(D ) ≥ |(n − 4)Q + (n − 2)Q + (n + 2)Q + nQ |. r,n 2(n − 1) r+n+1 r+n r+2 r+1

4 Determinant and Inverse of Skew Left Circulant Matrix with the Pell-Lucas Numbers

′ In this section, let Dr,n := SLCirc(Qr+1,Qr+2,...,Qr+n), where Qi be the i-th Pell-Lucas number ′ for i = r + 1, . . . , r + n. Firstly, we give a determinant explicit formula for the matrix Dr,n. ′ Afterwards, we prove that Dr,n is an invertible matrix for any positive integer n, and then we ﬁnd ′ the inverse of the matrix Dr,n. By Lemma 1.2–1.3, Theorem 3.1–3.3, we can obtain the following theorem.

′ Theorem 4.1. Let Dr,n := SLCirc(Qr+1,Qr+2,...,Qr+n). Then we have

′ (i).Dr,n is invertible for any positive integer n, n∑−2 ′ n(n−1) − 2 n (k+1) (ii). det Dr,n = (−1) Qr+1 · [Qr+1 + θQr+n + (θQr+k+1 − Qr+k+2)·y ] k=1 n−2 · (Qr+1 + Qr+n+1) ,

′−1 1 ′′ ′′ ′′ (iii).D r,n = (y1 , y2 , . . . , yn ), qn where n∑−3 − − ′′ (−d)i 1 (−d)n 3 y = 1 − (Q − − θQ − )· − 2(Q − θQ ) · , 1 r+n+2 i r+n+1 i ci r+3 r+2 cn−2 i=1 ∑2 − ′′ (−d)n 5+i y = (Q − θQ )· , 2 r+1+i r+i cn−4+i i=1 ∑2 − − ′′ (−d)n 3 k+i y = (Q − θQ )· , (k = 3, 4, . . . , n − 3), k r+1+i r+i cn−2−k+i i=1 ∑2 − ′′ (−d)i 1 y − = (Q − θQ ) · , n 2 r+1+i r+i ci i=1 ′′ 1 y − = (Q − θQ ) · , n 1 r+3 r+2 c n∑−2 − ′′ (−d)i 1 y = θ + (Q − − Q − ) · . n r+n+1 i r+n i ci i=1 5 Norm and Spread of Skew Left Circulant Matrix with the Pell Lucas Numbers

′ ′ Theorem 5.1. Let Dr,n := SLCirc(Qr+1,Qr+2, ..., Qr+n)(n > 2). Then three kinds norms of Dr,n are given by

′ ′ 1 ∥D ∥ = ∥D ∥∞ = (Q + Q − Q − Q ), r,n 1 r,n 2 r+n+1 r+n r+1 r

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√ ′ n ∥D ∥ = (Q Q − Q Q ). (5.1) r,n F 2 r+n+1 r+n r r+1

′′ Theorem 5.2. Let Dr,n := SLCirc(Qr+1, −Qr+2, ..., −Qr+n−1,Qr+n) and n be odd. Then

∑n ′′ 1 ∥D ∥ = Q = (Q + Q − Q − Q ). r,n 2 r+i 2 r+n+1 r+n r+1 r i=1 Proof. According to Lemma 1.2, we have

n ′′ ∑ i−1 (j− 1 )(k−1) n − 1 λ (D ) = ±| (−1) Q ω 2 |, (j = 1, 2,..., ), j r,n r+i 2 i=1 and n ′′ ∑ λ n+1 (Dr,n) = Qr+i, (5.2) 2 i=1 thus

n ′′ ∑ i−1 i−1 |λj (Dr,n)| ≤ |(−1) Qr+i(−1) | i=1 n ∑ n + 1 = Q , (j = 1, 2,..., ). (5.3) r+i 2 i=1 By Equation (5.2) and (5.3), we have

∑n ′′ max λi(Dr,n) = Qr+i. (5.4) 0≤i≤ n+1 2 i=1 By Lemma 1.6, Equation (1.1) and Equation (5.3), we obtain

∑n ′′ 1 ∥D ∥ = Q = (Q + Q − Q − Q ). r,n 2 r+i 2 r+n r+n+1 r+1 r i=1

′ ′ Theorem 5.3. Let Dr,n := SLCirc(Qr+1,Qr+2, ..., Qr+n) . Then the bound for the spread of Dr,n satisﬁes:

(i) if n is odd, then √ ′ 1 2Q ≤s(D ) ≤ nM − N 2, r+n r,n 2n where

M = Qr+nQr+n+1 − QrQr+1,

N = Qr+n + Qr+n−1 + Qr+1 − Qr;

(ii) if n is even, then √ ′ 2Qr+n ≤s(Dr,n) ≤ n(Qr+nQr+n+1 − QrQr+1).

13 Yao and Sun; JAMCS, 26(2): 1-16, 2018; Article no.JAMCS.38768

′ ′ Proof. Since Dr,n is a symmetric matrix, by Lemma 1.7, s(Dr,n) ≥ 2 max |aij | = 2Qr+n, in addition, i≠ j

(i) if n is odd, ′ tr(Cr,n) = Qr+1 − Qr+2 + Qr+3 − · · · + Qr+n

= Qr+1 + Qr+1 + Qr+2 + ··· + Qr+n−1 n∑−1 = Qr+1 + Qr+i. i=1 By Equation (1.1) we have

′ 1 tr(D ) = (Q + Q − + Q − Q ). (5.5) r,n 2 r+n r+n 1 r+1 r

Let

M = Qr+nQr+n+1 − QrQr+1,

N = Qr+n + Qr+n−1 + Qr+1 − Qr.

By Equation (1.10), (5.1) and (5.5), we obtain √ ′ 1 2Q ≤s(C ) ≤ nM − N 2. r+n r,n 2n (ii) If n is even, ′ tr(Dr,n) = Qr+1 − Qr+1 + Qr+3 − Qr+3 + ··· + Qr+n−1 − Qr+n−1 = 0. (5.6)

By Equation (1.10), (5.1) and (5.6), we obtain √ ′ 2Qr+n ≤s(Dr,n) ≤ n(Qr+nQr+n+1 − QrQr+1).

6 Conclusion

The determinants and inverses of skew circulant and skew left circulant matrices with Pell and Pell-Lucas numbers is discussed in this paper, and four kinds norms and bounds for the spread of these matrices are given, respectively.

Acknowledgement

This project is supported by the National Natural Science Foundation of China (Grant No. 11671187), Natural Science Foundation of Shandong Province (Grant No. ZR2016AM14) and the AMEP of Linyi University, China.

Competing Interests

Authors have declared that no competing interests exist.

14 Yao and Sun; JAMCS, 26(2): 1-16, 2018; Article no.JAMCS.38768

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