# VanderLaan Circulant Type Matrices

Authors: Hongyan Pan; Zhaolin Jiang
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, andg-circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaan g-circulant matrix.


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## Body

## 1. Introduction

It is well known that the circulant matrices are one of the most important research tools in control methods for modern complex systems. There are many results on circulant systems. The concept of “chart” was used to investigate structural controllability and fixed models in [1] while more attention was paid to properties and controller design for a special class of circulant systems, called symmetrically circulant systems [2–5].Systems with block symmetric circulant structure are a kind of complex systems and constitute an important class of large-scale systems where there may be a clear advantage of using multivariable control and where it is actually used in practice. Typical application examples can be found in the control of paper machine [6–8] and control of power systems with parallel structure [7]. Other examples, including multizone crystal growth furnaces and dyes for plastic films, were also listed in [7, 8] and the references therein. For those symmetric circulant composite systems, due to the high dimensionality of the overall system and information structural constraints, the control problem is more complex and few results on regional pole assignment are available in the literature. In [9], the authors considered the problem of placing the poles of uncertain symmetric circulant composite systems in a specified disk, which is also presented as the problem of quadratic D stabilisation. Lee et al. [10] presented linear quadratic (LQ) repetitive control (RC) methods for processes represented by a conventional FIR model and a circulant FIR model. The latter, which represents a FIR system under the assumption of a cyclic steady state, is named its input-output map. The map is represented by a circulant matrix. Using the complete frequency resolving property of a circulant matrix, a special tuning method for the LQ weights is proposed. Lee and Won considered properties of pulse response circulant matrix and applied that to MIMO control and identification in [11].Furthermore, circulant type matrices have been put on the firm basis with the work in [12–18] and so on. These special matrices have significant applications in various disciplines.The VanderLaan sequences are defined by the following recurrence relation [19]:
(1)Vn=Vn-2+Vn-3,
where
(2)V0=0,V1=1,V2=0.For the convenience of readers, we gave the first few values of the sequences as follows:(3)n012345678⋯Vn010111223⋯.The characteristic equation of VanderLaan numbers isx3-x-1=0, and its roots are denoted by r1, r2, r3; then
(4)r1+r2+r3=0,r1r2+r1r3+r2r3=-1,r1r2r3=1.
The Binet form for VanderLaan sequence is
(5)Vn=c1r1n+c2r2n+c3r3n,
where
(6)c1=r2+r3(r2-r1)(r1-r3),c2=r1+r3(r2-r1)(r3-r2),c3=r1+r2(r2-r3)(r3-r1).Some authors have presented the explicit determinants and inverse of circulant type matrices involving famous numbers in recent years. For example, in [14], the nonsingularity of circulant type matrices with the sum and product of Fibonacci and Lucas numbers is discussed. And the exact determinants and inverses of these matrices are given. Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers are given in [20]. Jiang et al. [15] studied circulant type matrices with the k-Fibonacci and k-Lucas numbers and obtained the explicit determinants and inverse matrices. Lin [21] proposed the determinants of the Fibonacci-Lucas quasi-cyclic matrices. In [22], Jiang et al. discussed the nonsingularity of the skew circulant type matrices and presented explicit determinants and inverse matrices of these special matrices. Furthermore, four kinds of norms and bounds for the spread of these matrices are given separately. In [23], Shen et al. discussed circulant matrices with Fibonacci and Lucas numbers and gave their explicit determinants and inverses. Authors [24] discussed the nonsingularity of the circulant matrix and presented the explicit determinant and inverse matrices. Moreover, the nonsingularity of the left circulant and g-circulant matrices is also studied. The explicit determinants and inverse matrices of the left circulant and g-circulant matrices are obtained by utilizing the relationship between left circulant and g-circulant matrices and circulant matrix, respectively.This paper is aimed at getting the more beautiful results for the determinants and inverses of circulant type matrices via some surprising properties of VanderLaan numbers.A VanderLaan circulant matrix is ann×n matrix of the following form:
(7)Circ(Vr+1,Vr+2,…,Vr+n)=Vr+1Vr+2…Vr+nVr+nVr+1…Vr+n-1⋮⋮⋮Vr+2Vr+3…Vr+1.A VanderLaan left circulant matrix is ann×n matrix of the following form:
(8)LCirc(Vr+1,Vr+2,…,Vr+n)=Vr+1Vr+2…Vr+nVr+2Vr+3…Vr+1⋮⋮⋮Vr+nVr+1…Vr+n-1.A VanderLaang-circulant matrix is an n×n matrix with the following form:
(9)Ag,n=Vr+1Vr+2…Vr+nVr+n-g+1Vr+n-g+2…Vr+n-gVr+n-2g+1Vr+n-2g+2…Vr+n-2g⋮⋮⋱⋮Vr+g+1Vr+g+2…Vr+g,
where g is a nonnegative integer and each of the subscripts is understood to be reduced modulo n.Lemma 1.
Define then×n matrix by
(10)An=κ1κ2⋯00κ3κ1⋯000κ3⋯00⋯⋯⋱⋯⋯00⋯κ1κ200⋯κ3κ1;
then determinant of the matrix An is
(11)det⁡An=κ1+κ12-4κ2κ32n+1-κ1-κ12-4κ2κ32n+1×κ12-4κ2κ3-1,κ12≠4κ2κ3,(n+1)κ12n,κ12=4κ2κ3.Proof.
A calculation using the expansion of the last column for determinant of matrixAn shows that det⁡An=κ1·det⁡An-1-κ2κ3·det⁡An-2; let x+y=κ1, xy=κ2κ3; then let x, y be the roots of the equation x2-κ1x+κ2κ3=0.
We have(12)det⁡An=yn+xyn-1+⋯+xn-1y+xn=xn+1-yn+1x-y,x≠y,(n+1)xn,x=y,
where
(13)x=κ1+κ12-4κ2κ32,y=κ1-κ12-4κ2κ32.
We obtain
(14)det⁡An=κ1+κ12-4κ2κ32n+1-κ1-κ12-4κ2κ32n+1×κ12-4κ2κ3-1,κ12≠4κ2κ3,(n+1)κ12n,κ12=4κ2κ3.Lemma 2.
Let(15)Bn=a1a2⋯an-1anκ1κ2⋯00κ3κ1⋯00⋯⋯⋱⋯⋯00⋯κ2000⋯κ1κ2
be a  n×n matrix; then
(16)det⁡Bn=∑i=1n‍-11+iκ2n-iai·det⁡Ai-1,
where, for n≥3,
(17)det⁡Ai-1=κ1+κ12-4κ2κ32i-κ1-κ12-4κ2κ32i×κ12-4κ2κ3-1,κ12≠4κ2κ3,iκ12i-1,κ12=4κ2κ3,
and det⁡A0=1.Proof.
Expanding the last column for the determinant of matrixBn and according to Lemma 1, we obtain
(18)det⁡Bn=κ2·det⁡Bn-1+-1n+1an·det⁡An-1=κ2n-1a1+-13κ2n-2a2·det⁡A1+⋯+-1n+1an·det⁡An-1=∑i=1n‍-11+iκ2n-iai·det⁡Ai-1.
This completes the proof.Lemma 3.
LetΦ=aBCA be a partitioned matrix; then
(19)Φ-1=1l-1lBA-1-1lA-1CA-1+1lA-1CBA-1,
where l=a-BA-1C, B is a row vector, and C is a column vector.Proof.
From direct calculation by matrix multiplication, we get(20)ΦΦ-1=In,Φ-1Φ=In,
where
(21)Φ=aBCA,Φ-1=1l-1lBA-1-1lA-1CA-1+1lA-1CBA-1.Lemma 4.
Let the matrixG=hi,ji,j=1n-3 be of the form
(22)hi,j=c,i=j,b,i=j+1,d,i=j+2,0,otherwise;
then the inverse G-1=hi,j′i,j=1n-3 of the matrix G is equal to
(23)hi,j′=1cβi-j+1-αi-j+1β-α,i≥j,0,i<j,
where
(24)c=Vr+1-Vr+n+1,b=Vr+2-Vr+n-Vr+n-1,d=Vr-Vr+n,α=-b+b2-4cd2c,β=-b-b2-4cd2c.Proof.
Letci,j′=∑k=1n-3‍hi,khk,j′; if i=j, we have ci,i′=hi,ihi,i′=1; distinctly, ci,j′=0 for i<j. And, for i≥j+1, we have
(25)ci,j′=hi,i-2hi-2,j′+hi,i-1hi-1,j′+hi,ihi,j′=dcβi-j-1-αi-j-1β-α+βi-j+1-αi-j+1β-α+bcβi-j-αi-jβ-α=0.
We obtain GG-1=In-3, where In-3 is (n-3)×(n-3) identity matrix. Evidenced by the same token, G-1G=In-3. The proof is completed.We first introduce the following notations:(26)ν=e+∑i=2n-1‍δi+1Δn-i,Δ=-b±b2-4cd2c,e=Vr+1-Vr+2Vr+nVr+1,δi+1=Vr+i+1-Vr+2Vr+iVr+1i=2,3,…,n-1,κ=∑i=2n-2‍-1i+1cn-i-2ϵn-iϕi-1+ϵ1cn-3,ϵ1=Vr+1-Vr+3Vr+n-1Vr+1,ϵi=Vr+i+2-Vr+3Vr+iVr+1(i=2,3,…,n-2),ϕi-1=b+b2-4cd2i-b-b2-4cd2i×b2-4cd-1,b2≠4cd,ib2i-1,b2=4cd,θ=f+∑i=2n-2‍ϵiΔn-i,f=Vr+2-Vr+3Vr+nVr+1+ϵ1Δ,τ=∑i=1n-2‍-1i+1cn-i-2δn-i+1ϕi-1,p=Vr+2-Vr+n+2.


## 2. Determinant and Inverse of VanderLaan Circulant Matrix

In this segment, letVr,n be a VanderLaan circulant matrix. To begin with, we give the determinant of the matrix Vr,n. And then we draw a conclusion that Vr,n is an invertible matrix for n≠2kπ/arctan4cd-p2/±p; finally we find the inverse of the matrix Vr,n.Theorem 5.
LetVr,n=Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan circulant matrix, so its determinant is
(27)det⁡Vr,n=Vr+1(νκ-θτ),
where Vr+n is the (r+n)th VanderLaan number.Proof.
Apparently,det⁡V0,4=3 meets formula (27). If n>4, let
(28)Γ1=1-Vr+2Vr+11-Vr+3Vr+100-1⋰-10⋰-1⋮⋰⋰⋰⋰⋮010-1-1⋯0,Π1=100⋯000Δn-20⋯010Δn-30⋯10⋮⋮⋮⋰⋮⋮0Δ1⋯00010⋯00
be two n×n matrices; we have
(29)Γ1Vr,nΠ1=Vr+1μVr+n-1⋯Vr+20νδn⋯δ30θϵ1⋯ϵ200b⋯000d⋯0⋮⋮⋮⋮000⋯c,
where
(30)μ=∑i=2n‍Vr+iΔn-i;
we obtain
(31)det⁡Γ1det⁡Vr,ndet⁡Π1=Vr+1·(νκ-θτ).
Let
(32)Bn-2=ϵ1ϵn-2ϵn-3⋯ϵ2bc0⋯0dbc⋯0⋮⋮⋮⋯⋮000⋯0000⋯c,Cn-2=δnδn-1δn-2⋯δ3bc0⋯0dbc⋯0⋮⋮⋮⋯⋮000⋯0000⋯c
be two (n-2)×(n-2) matrices, and
(33)κ=det⁡Bn-2,τ=det⁡Cn-2.
By Lemmas 1 and 2, the following equations hold:
(34)τ=∑i=1n-2‍-1i+1cn-i-2δn-i+1ϕi-1,κ=∑i=2n-2‍-1i+1cn-i-2ϵn-iϕi-1+ϵ1cn-3;
while
(35)det⁡Γ1=det⁡Π1=-1(n-1)(n-2)/2,
we have
(36)det⁡Vr,n=Vr+1νκ-θτ,
which completes the proof.Theorem 6.
LetVr,n=Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan circulant matrix; in the case of n≠2kπ/arctan⁡(4cd-p2/±p), Vr,n is a nonsingular matrix.Proof.
According to Theorem5, in case of n≤4, we have det⁡Vr,n≠0. When n>4, since Vn=c1r1n+c2r2n+c3r3n, where
(37)c1=r2+r3(r2-r1)(r1-r3),c2=r1+r3(r2-r1)(r3-r2),c3=r1+r2(r2-r3)(r3-r1),
we get
(38)fωk=∑j=1n‍Vr+jωkj-1=∑j=1n‍c1r1r+j+c2r2r+j+c3r3r+jωkj-1=dω2k+Vr+2-Vr+n+2ωk+cσ,
where
(39)σ=1-r1ωk1-r2ωk1-r3ωk,(k=1,…,n-1).
If there exists  ωl(l=1,2,…,n-1) showing that f(ωl)=0, here we get dω2k+pωk+c=0 for (1-r1ωk)(1-r2ωk)(1-r3ωk)≠0. If p2-4cd≥0, thus, ωl=(-p±p2-4cd)/2 is a real number. While
(40)ωl=exp⁡2lπin=cos⁡2lπn+isin2lπn,sin(2lπ/n)=0, we obtain ωl=-1 for 0<2lπ/n<2π. But x=-1 is not the root of the equation dx2+px+c=0. We have f(ωk)≠0 for any ωk(k=1,2,…,n-1), while f(1)=∑j=1n‍Vr+j=((Vr+Vr+4-Vr+n-Vr+n+4)/((1-r1)(1-r2)(1-r3)))≠0. If p2-4cd<0, we get that ωk is an imaginary number, if and only if
(41)cos⁡2kπn=-p2d,sin2lπn=±4cd-p22d.
We obtain n=2kπ/arctan⁡(4cd-p2/±p), such that f(ωk)=0. If (1-r1ωk)(1-r2ωk)(1-r3ωk)=1-ω2k-ω3k=0, we have ωk=1/r1, 1/r2, or 1/r3; obviously, ωk≠0 and ωk≠±1. In the same way, we know that 1/ri≠0 and 1/ri≠±1. So, f(1/ri)=∑j=1n‍Vr+j(1/ri)j-1≠0, (i=1,2,3). By Lemma 1 in [14], the proof is obtained.Theorem 7.
LetVr,n=Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan circulant matrix. If n≠2kπ/arctan⁡(4cd-p2/±p), so its inverse is
(42)Vr,n-1=Circ1ν-θνy3′-y4′-y5′,-Vr+2νVr+1+θVr+2νVr+1-Vr+3Vr+1y3′-y4′,yn′,…,y3′-y5′-y6′1ν-θνy3′-y4′-y5′,-Vr+2νVr+1+θVr+2νVr+1-Vr+3Vr+1y3′,
where
(43)y1′=0,y2′=1ν,y3′=b3ξ-c∑i=2n-2‍hi1bi+2ξ,y4′=-b3∑i=1n-3‍hi1ρi+3ξ+∑i=1n-3‍hi1bi+3-c∑i=1n-3‍hi1ρi+3∑i=2n-2‍hi1bi+2ξ,⋮yk′=-b3∑i=1n-k+1‍hi1ρi+k-1ξ+∑i=1n-k+1‍hi1bi+k-1-c∑i=1n-k+1‍hi1ρi+k-1∑i=2n-2‍hi1bi+2ξ,k≥4,ξ=ρ3-b∑i=1n-3‍hi1ρi+3-d∑i=1n-4‍hi1ρi+4,ρ3=Vr+1-Vr+3Vr+n-1Vr+1-θνδn,ρi=Vr+n-i+4-θνδn-i+3-Vr+3Vr+n+2-iVr+1(i=4,…,n),bj=Vr+2Vr+n+2-j-Vr+1Vr+n+3-jνVr+1(j=3,…,n).Proof.
Let(44)Γ2=100⋯0010⋯00-θν1⋯0⋯⋯⋯⋱⋯000⋯1;
thus
(45)ΓVnΠ1=Vr+1μVr+n-1⋯Vr+20νδn⋯δ300ρ3⋯ρn00b⋯0⋯⋯⋯⋯000⋯c.
Also we have Γ=Γ2Γ1, according to Lemma 3; letting
(46)Ψ=ρ3VUG,
be a (n-2)×(n-2) partitioned matrix, we obtain
(47)Ψ-1=1ξ-VG-1ξ-G-1UξG-1+UVG-1ξ,
where
(48)ξ=ρ3-VG-1U,U=b,d,0,…,0T,V=(ρ4,ρ5,…,ρn).
Let
(49)Π2=1-μVr+1ϱ3⋯ϱn01b3⋯bn001⋯0⋮⋮⋮⋱⋮000⋯1,
where
(50)ϱi=μbi-Vr+n-i+2Vr+1(i=3,…,n).
So
(51)ΓVr,nΠ1Π2=D⊕Ψ,
where D=diag⁡(Vr+1,ν) is a diagonal matrix and D⊕Ψ is the direct sum of D and Ψ. If we denote Π=Π1Π2, then we obtain
(52)Vr,n-1=Π(D-1⊕Ψ-1)Γ,Γ=100-Vr+2Vr+1001χ⋰1-θν-1⋰⋰0-10⋰-1-1⋮⋰⋰⋰⋰⋰00010-1⋰0⋮010-1-10⋯0,
where
(53)χ=θVr+2νVr+1-Vr+3Vr+1.
Since the last row elements of the matrix Π are 0,1,(Vr+2Vr+n-1-Vr+1Vr+n)/νVr+1,…,(Vr+22-Vr+1Vr+3)/νVr+1, then for Π(D1-1⊕Ψ-1) its last row elements are given by the following equations:
(54)y1′=0,y2′=1ν,y3′=b3ξ-c∑i=2n-2‍hi1bi+2ξ,y4′=-b3∑i=1n-3‍hi1ρi+3ξ+∑i=1n-3‍hi1bi+3-c∑i=1n-3‍hi1ρi+3∑i=2n-2‍hi1bi+2ξ,⋮yk′=-b3∑i=1n-k+1‍hi1ρi+k-1ξ+∑i=1n-k+1‍hi1bi+k-1-c∑i=1n-k+1‍hi1ρi+k-1∑i=2n-2‍hi1bi+2ξ,(k≥4).
By Lemma4, if Vr,n-1=Circ(y1,y2,…,yn), we have its last row elements as follows:
(55)y2=-Vr+2νVr+1+θVr+2νVr+1-Vr+3Vr+1y3′-y4′,y3=yn′,y4=yn-1′,y5=yn-2′-yn′,⋮yk=yn-k+3′-yn-k+5′-yn-k+6′(5<k≤n),y1=1ν-θνy3′-y4′-y5′.
Proof is completed.


## 3. Norms and Bound of Spread for VanderLaan Circulant Matrix

In this section, we gain three kinds of norms and lower bound for the spread of VanderLaan circulant matrix separately.Theorem 8.
LetVr,n=Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan circulant matrix. One has its two kinds of norms:
(56)Vr,n1=Vr,n∞=Vr+n+5-Vr+5.Proof.
By definition of norms in [25] and (4), we obtain
(57)Vr,n1=Vr,n∞=∑j=1n‍Vr+j=Vr+Vr+1+Vr+2-Vr+n-Vr+n+1-Vr+n+2(1-r1)(1-r2)(1-r3)=Vr+5-Vr+n+51-r11-r21-r3=Vr+5-Vr+n+51-r1+r2+r3+r1r2+r2r3+r1r3-r1r2r3=Vr+n+5-Vr+5.
Theorem 8 is proved.Theorem 9.
LetVr,n=Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan circulant matrix. So one has
(58)Vr,n2=∑j=1n‍Vr+j=Vr+n+5-Vr+5.Proof.
According to Theorem 2 in [26] and (4), we get
(59)Vr,n2=∑j=1n‍Vr+j=Vr+Vr+1+Vr+2-Vr+n-Vr+n+1-Vr+n+2(1-r1)(1-r2)(1-r3)=Vr+5-Vr+n+51-(r1+r2+r3)+(r1r2+r2r3+r1r3)-r1r2r3=Vr+n+5-Vr+5.
The proofs are completed.Theorem 10.
LetVr,n=Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan circulant matrix; the lower bound for the spread of Vr,n is
(60)s(Vr,n)≥n(Vr+n+5-Vr+6)n-1.Proof.
By (19) in [27] and (4), we obtain
(61)sVr,n≥nn-1∑j=2n‍Vr+j.
Since
(62)∑j=2n‍Vr+j=c1r1r+2(1-r1n-1)(1-r1)+c2r2r+2(1-r2n-1)(1-r2)+c3r3r+2(1-r3n-1)(1-r3)=Vr+1+Vr+2+Vr+3-Vr+n-Vr+n+1-Vr+n+21-r11-r21-r3=Vr+6-Vr+n+51-r1+r2+r3+r1r2+r2r3+r1r3-r1r2r3=Vr+n+5-Vr+6,
we get
(63)sVr,n≥nVr+n+5-Vr+6n-1,
which completes the proof.


## 4. Determinant, Inverse, and Norms and Spread of VanderLaan Left Circulant Matrix

In this part, letUr,n be a VanderLaan left circulant matrix. By using the obtained conclusions, we give a determinant formula for the matrix Ur,n. Afterwards, we prove that Ur,n is an nonsingular matrix for n≠2kπ/arctan⁡4cd-p2/±p. The inverse of the matrix Ur,n is also presented. Finally, three kinds of norms and lower bound for the spread of VanderLaan left circulant matrix are given.According to Lemma 2 in [15] and Theorems 5, 6, and 7, we can obtain the following theorems.Theorem 11.
LetUr,n=LCirc(Vr+1,Vr+2,…,Vr+n) be a VanderLaan left circulant matrix; then one has
(64)det⁡Ur,n=-1(n-1)(n-2)/2[Vr+1(νκ-θτ)],
where Vr+n is the (r+n)th VanderLaan number.Theorem 12.
LetUr,n=LCirc(Vr+1,Vr+2,…,Vr+n) be a VanderLaan left circulant matrix; if n≠2kπ/arctan⁡(4cd-p2/±p), then Ur,n is a nonsingular matrix.Theorem 13.
LetUr,n=LCirc(Vr+1,Vr+2,…,Vr+n) be a VanderLaan left circulant matrix; in case of n≠2kπ/arctan⁡(4cd-p2/±p), its inverse is
(65)Ur,n-1=LCirc1ν-θνy3′-y4′-y5′,y3′-y5′-y6′,…,yn′,-Vr+2νVr+1+(θVr+2νVr+1-Vr+3Vr+1)y3′-y4′,
where
(66)y1′=0,y2′=1ν,y3′=b3ξ-c∑i=2n-2‍hi1bi+2ξ,y4′=-b3∑i=1n-3‍hi1ρi+3ξ+∑i=1n-3‍hi1bi+3-c∑i=1n-3‍hi1ρi+3∑i=2n-2‍hi1bi+2ξ,⋮yk′=-b3∑i=1n-k+1‍hi1ρi+k-1ξ+∑i=1n-k+1‍hi1bi+k-1-c∑i=1n-k+1‍hi1ρi+k-1∑i=2n-2‍hi1bi+2ξ,(k≥4).Theorem 14.
LetUr,n=LCirc(Vr+1,Vr+2,…,Vr+n) be a VanderLaan left circulant matrix, so one can get two kinds of norms of Ur,n:
(67)Ur,n1=Ur,n∞=Vr+n+5-Vr+5.Proof.
By definition of norms in [25] and (57), we have
(68)Ur,n1=Ur,n∞=∑j=1n‍Vr+j=Vr+n+5-Vr+5.
This completes the proof.Theorem 15.
LetUr,n=LCirc(Vr+1,Vr+2,…,Vr+n) be a VanderLaan left circulant matrix; then one has the spectral norm of Ur,n:
(69)Ur,n2=Vr+n+5-Vr+5.Proof.
From Theorem 2 in [26] and (57), we can have
(70)Ur,n2=∑j=1n‍Vr+j=Vr+n+5-Vr+5.
The proofs are completed.Theorem 16.
LetUr,n=LCirc(Vr+1,Vr+2,…,Vr+n) be a VanderLaan left circulant matrix; the lower bound for the spread of Ur,n is
(71)s(Ur,n)≥Vr+n+5-Vr+5,nisodd,t,niseven,
where
(72)t=nn-1(Vr+n+5-Vr+5)-2n-1(Vr+n+2-Vr+2).Proof.
By (19) in [27], we obtain
(73)sUr,n≥1n-1∑j≠i‍Vr+j.
When n is odd
(74)1n-1∑j≠i‍Vr+j=1n-1[n∑j=1n‍Vr+j-∑j=1n‍Vr+j]=∑j=1n‍Vr+j=Vr+n+5-Vr+5.
If n is even
(75)1n-1∑j≠i‍Vr+j=1n-1[n∑j=1n‍Vr+j-2∑j=1n/2‍Vr+2j-1]=nn-1(Vr+n+5-Vr+5)-2n-1∑j=1n/2‍Vr+2j-1=nn-1Vr+n+5-Vr+5-2n-1Vr+n+2-Vr+2=t.
We get
(76)s(Ur,n)≥Vr+n+5-Vr+5,nisodd,t,niseven.
The proofs are completed.


## 5. Determinant, Inverse, and Spectral Norm of VanderLaang-Circulant Matrix

In this section, letWg,r,n be a VanderLaan g-circulant matrix. We give a determinant formula for the matrix Wg,r,n by the means of the gained results. Afterwards, we get the inverse of the matrix Wg,r,n and obtain that Wg,r,n is an invertible matrix for n≠2kπ/arctan⁡(4cd-p2/±p). At last, we gain the spectral norm of VanderLaan g-circulant matrix.From Lemmas 3 and 4 in [15] and Theorems 5, 6, and 7, the following results are deduced.Theorem 17.
LetWg,r,n=g-Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan g-circulant matrix and (n,g)=1; one obtains
(77)det⁡Wg,r,n=Vr+1(νκ-θτ)det⁡Qg,
where Vr+n is the (r+n)th VanderLaan number and the matrix Qg is given Lemma 3 in [15].Theorem 18.
LetWg,r,n=g-Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan g-circulant matrix and (n,g)=1; if n≠2kπ/arctan⁡(4cd-p2/±p), Wg,r,n is a nonsingular matrix.Theorem 19.
LetWg,r,n=g-Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan g-circulant matrix and (n,g)=1; when n≠2kπ/arctan⁡(4cd-p2/±p), the inverse of matrix Wg,r,n is
(78)Wg,r,n-1=Circ1ν-θνy3′-y4′-y5′,-Vr+2νVr+1+θVr+2νVr+1-Vr+3Vr+1y3′-y4′,yn′,…,y3′-y5′-y6′1ν-θνy3′-y4′-y5′,-Vr+2νVr+1Circ1ν-θνy3′-y4′-y5′,-Vr+2νVr+1QgT,
where
(79)y1′=0,y2′=1ν,y3′=b3ξ-c∑i=2n-2‍hi1bi+2ξ,y4′=-b3∑i=1n-3‍hi1ρi+3ξ+∑i=1n-3‍hi1bi+3-c∑i=1n-3‍hi1ρi+3∑i=2n-2‍hi1bi+2ξ,⋮yk′=-b3∑i=1n-k+1‍hi1ρi+k-1ξ+∑i=1n-k+1‍hi1bi+k-1-c∑i=1n-k+1‍hi1ρi+k-1∑i=2n-2‍hi1bi+2ξ,(k≥4).Theorem 20.
LetWg,r,n=g-Circ(Vr+1,Vr+2,…,Vr+n) be a VanderLaan g-circulant matrix and (n,g)=1. One has its spectral norm
(80)Wg,r,n2=Vr+n+5-Vr+5.Proof.
According to the definition of norms, Lemmas 3 and 4 in [15], and Theorem 9, if (g,n)=1, we get Wg,r,nTWg,r,n=Vr,nTQgTQgVr,n=Vr,nTVr,n; then ∥Wg,r,n∥2=∥Vr,n∥2=Vr+n+5-Vr+5.
This completes the proofs.


## 6. Conclusion

In this paper, we considered VanderLaan circulant type matrices. We discussed the nonsingularity of these special matrices and presented the exact determinants and inverse matrices of VanderLaan circulant type matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaang-circulant matrix.Furthermore, based on circulant matrices technology we will consider solving the problem in [28–31].



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*Source: 329329-2015-01-15.xml*