AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2015/329329 329329 Research Article VanderLaan Circulant Type Matrices Pan Hongyan http://orcid.org/0000-0002-9521-9835 Jiang Zhaolin Yin Shen 1 Department of Mathematics Linyi University Linyi, Shandong 276000 China lyu.edu.cn 2015 1512015 2015 15 08 2014 14 10 2014 1512015 2015 Copyright © 2015 Hongyan Pan and Zhaolin Jiang. 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 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, and g-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.

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  while more attention was paid to properties and controller design for a special class of circulant systems, called symmetrically circulant systems .

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  and control of power systems with parallel structure . 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 , 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.  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 .

Furthermore, circulant type matrices have been put on the firm basis with the work in  and so on. These special matrices have significant applications in various disciplines.

The VanderLaan sequences are defined by the following recurrence relation : (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)n012345678Vn010111223.

The characteristic equation of VanderLaan numbers is x3-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 , 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 . Jiang et al.  studied circulant type matrices with the k-Fibonacci and k-Lucas numbers and obtained the explicit determinants and inverse matrices. Lin  proposed the determinants of the Fibonacci-Lucas quasi-cyclic matrices. In , 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 , Shen et al. discussed circulant matrices with Fibonacci and Lucas numbers and gave their explicit determinants and inverses. Authors  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 an n×n matrix of the following form: (7)Circ(Vr+1,Vr+2,,Vr+n)=Vr+1Vr+2Vr+nVr+nVr+1Vr+n-1Vr+2Vr+3Vr+1.

A VanderLaan left circulant matrix is an n×n matrix of the following form: (8)LCirc(Vr+1,Vr+2,,Vr+n)=Vr+1Vr+2Vr+nVr+2Vr+3Vr+1Vr+nVr+1Vr+n-1.

A VanderLaan g-circulant matrix is an n×n matrix with the following form: (9)Ag,n=Vr+1Vr+2Vr+nVr+n-g+1Vr+n-g+2Vr+n-gVr+n-2g+1Vr+n-2g+2Vr+n-2gVr+g+1Vr+g+2Vr+g, where g is a nonnegative integer and each of the subscripts is understood to be reduced modulo n.

Lemma 1.

Define the n×n matrix by (10)An=κ1κ200κ3κ1000κ30000κ1κ200κ3κ1; then determinant of the matrix An is (11)detAn=κ1+κ12-4κ2κ32n+1-κ1-κ12-4κ2κ32n+1×κ12-4κ2κ3-1,κ124κ2κ3,(n+1)κ12n,κ12=4κ2κ3.

Proof.

A calculation using the expansion of the last column for determinant of matrix An shows that detAn=κ1·detAn-1-κ2κ3·detAn-2; let x+y=κ1, xy=κ2κ3; then let x, y be the roots of the equation x2-κ1x+κ2κ3=0.

We have (12)detAn=yn+xyn-1++xn-1y+xn=xn+1-yn+1x-y,xy,(n+1)xn,x=y, where (13)x=κ1+κ12-4κ2κ32,y=κ1-κ12-4κ2κ32. We obtain (14)detAn=κ1+κ12-4κ2κ32n+1-κ1-κ12-4κ2κ32n+1×κ12-4κ2κ3-1,κ124κ2κ3,(n+1)κ12n,κ12=4κ2κ3.

Lemma 2.

Let (15)Bn=a1a2an-1anκ1κ200κ3κ10000κ2000κ1κ2 be a  n×n matrix; then (16)detBn=i=1n-11+iκ2n-iai·detAi-1, where, for n3, (17)detAi-1=κ1+κ12-4κ2κ32i-κ1-κ12-4κ2κ32i×κ12-4κ2κ3-1,κ124κ2κ3,iκ12i-1,κ12=4κ2κ3, and detA0=1.

Proof.

Expanding the last column for the determinant of matrix Bn and according to Lemma 1, we obtain (18)detBn=κ2·detBn-1+-1n+1an·detAn-1=κ2n-1a1+-13κ2n-2a2·detA1++-1n+1an·detAn-1=i=1n-11+iκ2n-iai·detAi-1. This completes the proof.

Lemma 3.

Let Φ=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 matrix G=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,ji,j=1n-3 of the matrix G is equal to (23)hi,j=1cβi-j+1-αi-j+1β-α,ij,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.

Let ci,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 ij+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,b24cd,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, let Vr,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 n2kπ/arctan4cd-p2/±p; finally we find the inverse of the matrix Vr,n.

Theorem 5.

Let Vr,n=Circ(Vr+1,Vr+2,,Vr+n) be a VanderLaan circulant matrix, so its determinant is (27)detVr,n=Vr+1(νκ-θτ), where Vr+n is the (r+n)th VanderLaan number.

Proof.

Apparently, detV0,4=3 meets formula (27). If n>4, let (28)Γ1=1-Vr+2Vr+11-Vr+3Vr+100-1-10-1010-1-10,Π1=100000Δn-20010Δn-30100Δ10001000 be two n×n matrices; we have (29)Γ1Vr,nΠ1=Vr+1μVr+n-1Vr+20νδnδ30θϵ1ϵ200b000d0000c, where (30)μ=i=2nVr+iΔn-i; we obtain (31)detΓ1detVr,ndetΠ1=Vr+1·(νκ-θτ). Let (32)Bn-2=ϵ1ϵn-2ϵn-3ϵ2bc00dbc00000000c,Cn-2=δnδn-1δn-2δ3bc00dbc00000000c be two (n-2)×(n-2) matrices, and (33)κ=detBn-2,τ=detCn-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)detΓ1=detΠ1=-1(n-1)(n-2)/2, we have (36)detVr,n=Vr+1νκ-θτ, which completes the proof.

Theorem 6.

Let Vr,n=Circ(Vr+1,Vr+2,,Vr+n) be a VanderLaan circulant matrix; in the case of n2kπ/arctan(4cd-p2/±p), Vr,n is a nonsingular matrix.

Proof.

According to Theorem 5, in case of n4, we have detVr,n0. 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 p2-4cd0, thus, ωl=(-p±p2-4cd)/2 is a real number. While (40)ωl=exp2lπin=cos2lπn+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)cos2kπn=-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, ωk0 and ωk±1. In the same way, we know that 1/ri0 and 1/ri±1. So, f(1/ri)=j=1nVr+j(1/ri)j-10, (i=1,2,3). By Lemma 1 in , the proof is obtained.

Theorem 7.

Let Vr,n=Circ(Vr+1,Vr+2,,Vr+n) be a VanderLaan circulant matrix. If n2kπ/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-y61ν-θνy3-y4-y5,-Vr+2νVr+1+θVr+2νVr+1-Vr+3Vr+1y3, where (43)y1=0,y2=1ν,y3=b3ξ-ci=2n-2hi1bi+2ξ,y4=-b3i=1n-3hi1ρi+3ξ+i=1n-3hi1bi+3-ci=1n-3hi1ρi+3i=2n-2hi1bi+2ξ,yk=-b3i=1n-k+1hi1ρi+k-1ξ+i=1n-k+1hi1bi+k-1-ci=1n-k+1hi1ρi+k-1i=2n-2hi1bi+2ξ,k4,ξ=ρ3-bi=1n-3hi1ρi+3-di=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=100001000-θν100001; thus (45)ΓVnΠ1=Vr+1μVr+n-1Vr+20νδnδ300ρ3ρn00b0000c. 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+1ϱ3ϱn01b3bn00100001, where (50)ϱi=μbi-Vr+n-i+2Vr+1(i=3,,n). So (51)ΓVr,nΠ1Π2=DΨ, where D=diag(Vr+1,ν) is a diagonal matrix and DΨ is the direct sum of D and Ψ. If we denote Π=Π1Π2, then we obtain (52)Vr,n-1=Π(D-1Ψ-1)Γ,Γ=100-Vr+2Vr+1001χ1-θν-10-10-1-100010-10010-1-100, 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(D1-1Ψ-1) its last row elements are given by the following equations: (54)y1=0,y2=1ν,y3=b3ξ-ci=2n-2hi1bi+2ξ,y4=-b3i=1n-3hi1ρi+3ξ+i=1n-3hi1bi+3-ci=1n-3hi1ρi+3i=2n-2hi1bi+2ξ,yk=-b3i=1n-k+1hi1ρi+k-1ξ+i=1n-k+1hi1bi+k-1-ci=1n-k+1hi1ρi+k-1i=2n-2hi1bi+2ξ,(k4).

By Lemma 4, 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<kn),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.

Let Vr,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  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.

Let Vr,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  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.

Let Vr,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  and (4), we obtain (61)sVr,nnn-1j=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,nnVr+n+5-Vr+6n-1, which completes the proof.

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

In this part, let Ur,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 n2kπ/arctan4cd-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  and Theorems 5, 6, and 7, we can obtain the following theorems.

Theorem 11.

Let Ur,n=LCirc(Vr+1,Vr+2,,Vr+n) be a VanderLaan left circulant matrix; then one has (64)detUr,n=-1(n-1)(n-2)/2[Vr+1(νκ-θτ)], where Vr+n is the (r+n)th VanderLaan number.

Theorem 12.

Let Ur,n=LCirc(Vr+1,Vr+2,,Vr+n) be a VanderLaan left circulant matrix; if n2kπ/arctan(4cd-p2/±p), then Ur,n is a nonsingular matrix.

Theorem 13.

Let Ur,n=LCirc(Vr+1,Vr+2,,Vr+n) be a VanderLaan left circulant matrix; in case of n2kπ/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-ci=2n-2hi1bi+2ξ,y4=-b3i=1n-3hi1ρi+3ξ+i=1n-3hi1bi+3-ci=1n-3hi1ρi+3i=2n-2hi1bi+2ξ,yk=-b3i=1n-k+1hi1ρi+k-1ξ+i=1n-k+1hi1bi+k-1-ci=1n-k+1hi1ρi+k-1i=2n-2hi1bi+2ξ,(k4).

Theorem 14.

Let Ur,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  and (57), we have (68)Ur,n1=Ur,n=j=1nVr+j=Vr+n+5-Vr+5. This completes the proof.

Theorem 15.

Let Ur,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  and (57), we can have (70)Ur,n2=j=1nVr+j=Vr+n+5-Vr+5. The proofs are completed.

Theorem 16.

Let Ur,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 , we obtain (73)sUr,n1n-1jiVr+j. When n is odd (74)1n-1jiVr+j=1n-1[nj=1nVr+j-j=1nVr+j]=j=1nVr+j=Vr+n+5-Vr+5. If n is even (75)1n-1jiVr+j=1n-1[nj=1nVr+j-2j=1n/2Vr+2j-1]=nn-1(Vr+n+5-Vr+5)-2n-1j=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 VanderLaan <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M209"><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:math></inline-formula>-Circulant Matrix

In this section, let Wg,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 n2kπ/arctan(4cd-p2/±p). At last, we gain the spectral norm of VanderLaan g-circulant matrix.

From Lemmas 3 and 4 in  and Theorems 5, 6, and 7, the following results are deduced.

Theorem 17.

Let Wg,r,n=g-Circ(Vr+1,Vr+2,,Vr+n) be a VanderLaan g-circulant matrix and (n,g)=1; one obtains (77)detWg,r,n=Vr+1(νκ-θτ)detQg, where Vr+n is the (r+n)th VanderLaan number and the matrix Qg is given Lemma 3 in .

Theorem 18.

Let Wg,r,n=g-Circ(Vr+1,Vr+2,,Vr+n) be a VanderLaan g-circulant matrix and (n,g)=1; if n2kπ/arctan(4cd-p2/±p), Wg,r,n is a nonsingular matrix.

Theorem 19.

Let Wg,r,n=g-Circ(Vr+1,Vr+2,,Vr+n) be a VanderLaan g-circulant matrix and (n,g)=1; when n2kπ/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-y61ν-θνy3-y4-y5,-Vr+2νVr+1Circ1ν-θνy3-y4-y5,-Vr+2νVr+1QgT, where (79)y1=0,y2=1ν,y3=b3ξ-ci=2n-2hi1bi+2ξ,y4=-b3i=1n-3hi1ρi+3ξ+i=1n-3hi1bi+3-ci=1n-3hi1ρi+3i=2n-2hi1bi+2ξ,yk=-b3i=1n-k+1hi1ρi+k-1ξ+i=1n-k+1hi1bi+k-1-ci=1n-k+1hi1ρi+k-1i=2n-2hi1bi+2ξ,(k4).

Theorem 20.

Let Wg,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 , and Theorem 9, if (g,n)=1, we get Wg,r,nTWg,r,n=Vr,nTQgTQgVr,n=Vr,nTVr,n; then Wg,r,n2=Vr,n2=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 VanderLaan g-circulant matrix.

Furthermore, based on circulant matrices technology we will consider solving the problem in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and the AMEP of Linyi University, China.

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