The Determinants of Circulant and Skew-Circulant Matrices with Tribonacci Numbers

Mathematical Sciences And Applications E-Notes Volume 2 No. 2 pp. 67–75 (2014) c MSAEN

THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES WITH TRIBONACCI NUMBERS

DURMUS BOZKURT, CARLOS M. DA FONSECA AND FATIH YILMAZ

(Communicated by Irfan˙ S¸IAP)˙

Abstract. Determinant computation has an important role in mathematics. It can be computed by using some diﬀerent methods but it needs huge amount of operations to compute determinant of a matrix. For instance, using Gauss elimination method, it is neccessary about 2n3/3 arithmetic steps of a matrix of order n. Therefore, determinant computation has been considered for special matrices with special entries. Similarly, the permanent of a matrix is an analog of determinant where all the signs in the expansion by minors are taken as positive. This study considers the determinant of circulant matrices whose entries are Tribonacci numbers. Some relations with the permanent are established.

1. Introduction The Tribonacci sequence is given recursively as

Tn = Tn−1 + Tn−2 + Tn−3 , with T0 = 0 and T1 = T2 = 1, for n > 2 [8]. This is the sequence A000073 in the The On-Line Encyclopedia of Integer Sequences [11] and the ﬁrst few values are 1, 1, 2, 4, 7, 13, 24, 44, 81 ....

A circulant matrix of order n, Cn := circ (c0, c1, . . . , cn−1), associated with the numbers c0, c1, . . . , cn−1, is deﬁned as   c0 c1 . . . cn−2 cn−1  cn−1 c0 . . . cn−3 cn−2     ......  Cn =  . . . . .  .    c2 c3 . . . c0 c1  c1 c2 . . . cn−1 c0

Date: Received: June 27, 2014; Revised: July 24, 2014 Accepted: September 12, 2014. 2010 Mathematics Subject Classification. 11C20, 15A15, 11B39, 15B36. Key words and phrases. Circulant matrix; Tribonacci number; Determinant; Permanent. 67 68 DURMUS BOZKURT, CARLOS M. DA FONSECA AND FATIH YILMAZ where each row is a cyclic shift of the row above it [3]. The eigenvalues and eigenvectors of Cn are well known [13]: n−1 X jk λj = ckω , j = 0, 1, . . . , n − 1 , k=0 2πi √ where ω = exp( n ) and i = −1, and the corresponding eigenvectors are j 2j (n−1)j T xj = (1, ω , ω , . . . , ω ) , j = 0, 1, . . . , n − 1 . Therefore, we can write determinant of a nonsingular circulant matrix as: n−1 n−1 ! Y X jk det Cn = ckω j=0 k=0 where k = 0, 1, . . . , n − 1. A skew circulant matrix with the first row (a0, a1, ..., an−1) is meant to be a square matrix of the form   a0 a1 . . . an−2 an−1  −an−1 a0 . . . an−3 an−2     ......   . . . . .     −a2 −a3 . . . a0 a1  −a1 −a2 ... −an−1 a0 denoted by SCirc(a0, a1, ..., an−1) [7]. Circulant matrices have applications in many areas such as signal processing, coding theory, image processing and among others. Numerical solutions of elliptic and parabolic partial differential equations with periodic boundary conditions often involve linear systems associated with circulant matrices [2, 14]. Some special matrices involving well known number sequences have been recently investigated in different contexts. For example, Shen et al. [10] defined two circulant matrices whose elements are Fibonacci and Lucas numbers and derived formulas for determinants and inverses of the defined matrices. Similarly, Bozkurt and Tam [1] obtained determinant and inverse formulas for circulant matrices in terms of Ja- cobsthal and Jacobsthal-Lucas numbers. Alptekin et.al. [15] obtained some results for circulant and semi-circulant matrices. Moreover, there are many interesting relationships between determinants of matrices and such number sequences. For example, in [12] it is investigated the relationship between permanents of one type of Hessenberg matrix and the Pell and Perrin numbers. In this note, we considered circulant matrices whose entries are Tribonacci numbers. We obtain a formula for the determinant of these matrices. Then we analyze a family of matrices with permanents equal to the Tribonacci numbers. First, we will give a lemma to simplify the derivations. We remind that the Chebyshev poly- nomials of second kind satisfying {Un(x)}n>0, where each Un(x) is of degree n, satisfy the three-term recurrence relations

Un+1(x) = 2xUn(x) − Un−1(x) , for all n = 1, 2,..., with initial conditions U0(x) = 1 and U1(x) = 2x, or, equivalently, sin(n + 1)θ U (x) = , with x = cos θ (0 θ < π), n sin θ 6 THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES 69 for all n = 0, 1, 2 .... It is also standard (see e.g. [5]) that  a b   .. ..   c . .  √ n a det   = bc Un √ .  .. ..  2 bc  . . b  c a n×n Lemma 1.1. If   d1 d2 d3 ··· dn−1 dn  a b     c a b    (1.1) An =  ..   c a .     ......   . . .  c a b then n √ k−1 X n−k a (1.2) det An = dkb − bc Uk−1 √ , k=1 2 bc where Uk(x) is the kth Chebyshev polynomial of second kind. Proof. It is clear that √ n−1 n−1 a (1.3) det An = b det An−1 + (−) dn bc Un−1 √ . 2 bc

Applying now (1.3) recursively to det An−1, det An−2,..., det A1, one gets (1.2). 1.1. Determinant of Circulant matrices with Tribonacci numbers. In this section, we consider the n-square circulant matrix

Tn := circ (T1,T2,...,Tn) where Tn is the nth Tribonacci number. Then we give a formula for determinants of these matrices. First, let us deﬁne n-square matrix  1 0 0 ... 0 0   0 tn−2 0 ... 0 1   n−3   0 t 0 ... 1 0   n−4   0 t 0 0  (1.4) Qn =    ......   ......     0 t 1 ... 0 0  0 1 0 ... 0 0

2 here t is the positive root of the characteristic equation xnt + ynt + zn = 0, i.e., −y + py2 − 4x z t = n n n n 2xn where

xn = 1 − Tn+1, yn = −(Tn + Tn−1), and zn = −Tn . 70 DURMUS BOZKURT, CARLOS M. DA FONSECA AND FATIH YILMAZ

Then, let us consider n-square matrix Pn as below:  1 0 0 0 0 0 0 0 0 0 0   −1 0 0 0 0 0 0 0 0 0 1     −1 0 0 0 0 0 0 0 0 1 −1     −1 0 0 0 0 0 0 0 1 −1 −1     0 0 0 0 0 0 0 1 −1 −1 −1     0 0 0 0 0 0 1 −1 −1 −1 0  Pn =    0 0 0 0 0 1 −1 −1 −1 0 0     0 0 0 0 1 −1 −1 −1 0 0 0     . . . 0 0 0   ......   . . .   0 0 1 −1 −1 −1 0 0 . . .  0 1 −1 −1 −1 0 0 0 0 0 0 It can be seen that for all n > 3, 1, n ≡ 1 or 2 mod 4 det(P ) = det(Q ) = n n −1, n ≡ 0 or 3 mod 4. where Qn is deﬁned in (1.4) and det(PnQn) = 1. By matrix multiplication, we get;

(1.5) Sn = PnTnQn i.e.,   1 fn Tn−1 Tn−2 Tn−3 ··· T3 T2  0 gn Tn − Tn−1 Tn−1 − Tn−2 Tn−2 − Tn−3 ··· T4 − T3 T3 − T2     0 hn yn + 1 Tn−3 Tn−4 ··· T2 T1     0 0 yn xn  S =   n  0 0 zn yn xn     ......   . . . . .     0 0 zn yn xn  0 0 zn yn xn where n X n−i fn = − zit , i=2 n−1 X n−i gn = (Ti+1 − Ti)t + xn−1, i=2 n−2 X n−i hn = − zi−1t + (yn + 1)t + zn . i=2 A simple application of the Laplace expansion in (1.5), determinant will be

gn Tn − Tn−1 Tn−1 − Tn−2 Tn−2 − Tn−3 ··· T4 − T3 T3 − T2

hn yn + 1 Tn−3 Tn−4 ··· T2 T1

0 yn xn

0 zn yn xn |Sn| = ......

0 z y x n n n 0 zn yn xn THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES 71

hn Multiplying the ﬁrst row with −m = and adding it to the second row in Sn, gn we obtain

gn Tn−T n−1 Tn−1−T n−2 ··· T3−T 2

0 m(T −T n−1) + y +1 m(T −T n−2) + T ··· m(T −T 2) + T n n n−1 n−3 3 1 0 y x n n 0 zn yn xn |Sn| = ......

0 zn yn xn

0 zn yn xn Using Laplace expansion on the ﬁrst column, we get a Hessenberg matrix in the form of (1.1). Applying Lemma 1.1, we obtain for n > 3 n−1 det Tn = gn[m(Tn − Tn−1) + yn + 1]xn + n−2 X n−k √ k−1 yn + gn [m(Tn−k+1 − Tn−k) + Tn−k−1]xn (− xnzn) Uk−1 √ . 2 xnzn k=2 Determinant of Skew-Circulant matrices with Tribonacci numbers. In this section, we consider n-square skew circulant matrix

Kn := SCirc (T1,T2,...,Tn) here Tn denotes nth Tribonacci number. Then we give a formula for determinants of these matrices. In the ﬁrst step, let us deﬁne the n × n matrix  1 0 0 ... 0 0   0 mn−2 0 ... 0 1   n−3   0 m 0 ... 1 0   n−4   0 m 0 0  Mn =    ......   ......     0 m 1 ... 0 0  0 1 0 ... 0 0 2 here m is the positive root of the characteristic equation anm + bnm + cn = 0, i.e., −b + pb2 − 4a c m := n n n n 2an here an = 1 + Tn+1, bn = (Tn + Tn−1) and cn = Tn. Then consider n-square matrix Nn as below:  1 0 0 0 0 0 0 0 0 0 0   1 0 0 0 0 0 0 0 0 0 1     1 0 0 0 0 0 0 0 0 1 −1     1 0 0 0 0 0 0 0 1 −1 −1     0 0 0 0 0 0 0 1 −1 −1 −1     0 0 0 0 0 0 1 −1 −1 −1 0  Nn =   .  0 0 0 0 0 1 −1 −1 −1 0 0     0 0 0 0 1 −1 −1 −1 0 0 0     . . . 0 0 0   ......   . . .   0 0 1 −1 −1 −1 0 0 . . .  0 1 −1 −1 −1 0 0 0 0 0 0 72 DURMUS BOZKURT, CARLOS M. DA FONSECA AND FATIH YILMAZ

1, n ≡ 1 or 2 mod 4 It can be seen that det(M ) = det(N ) = for n > 3. n n −1, n ≡ 0 or 3 mod 4. Using matrix multiplication, we get

Wn := NnKnMn i.e., (1.6)  ∗  1 fn Tn Tn−1 Tn−2 ··· T3 T2 ∗  0 gn Tn−1 − Tn Tn−2 − Tn−1 Tn−3 − Tn−2 ··· T3 − T4 T2 − T3   ∗   0 hn bn + 1 −Tn−3 −Tn−4 · · · −T2 −T1     0 0 bn an  W =   n  0 0 cn bn an     ......   . . . . .     0 0 cn bn an  0 0 cn bn an here

n ∗ X n−i fn = Tim , i=2 n−1 ∗ X n−i gn = (Ti − Ti+1)m + Tn + 1, i=2 n−2 ∗ X n−i hn = − Ti−1m + (bn + 1)m + cn . i=2

Using Laplace expansion in (1.6),

∗ gn Tn−1 − Tn Tn−2 − Tn−1 Tn−3 − Tn−2 ··· T3 − T4 T2 − T3 ∗ hn bn + 1 −Tn−3 −Tn−4 · · · −T2 −T1

0 bn an

0 cn bn an |Wn| = ......

0 cn bn an

∗ hn Multiplying the ﬁrst row with s := − ∗ and adding it to the second row in Wn gn

∗ gn Tn−1 − Tn Tn−2 − Tn−1 ··· T3 − T4 T2 − T3

0 s(Tn−1 − Tn) + bn + 1 s(Tn−2 − Tn−1) − Tn−3 ··· s(T3 − T4) − T2 s(T2 − T3) − T1

0 bn an

|Wn| = . .. . cn bn .

. . 0 .. .. a n 0 cn bn an THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES 73 whose determinant is equal to

s(Tn−1 − Tn) + bn + 1 s(Tn−2 − Tn−1) − Tn−3 ··· s(T3 − T4) − T2 s(T2 − T3) − T1

bn an

∗ .. = g cn bn . . n ...... an

cn bn an Using Lemma 1.1, we get ∗ n−1 det Wn = gn [s(Tn−1 − Tn) + bn + 1]bn + n−2 ∗ X n−k p k−1 an + g [s(Tn−k − Tn−k+1) − Tn−k−1]b (− bncn) Uk−1 √ . n n 2 b c k=2 n n 1.1.1. Permanent and Tribonacci numbers. There are many references establishing determinantal representations of well-known number sequences. For example, in [9] Lee deﬁned the matrix  1 0 1   1 1 1     1 1 1    Ln =  .. ..   1 . .     .. ..   . . 1  1 1 and showed that per Ln = Ln−1, where Ln is the nth Lucas number. The authors [12] associated permanents of Hessenberg matrices with Pell and Perrin numbers. Recall that the permanent of a square matrix is similar to the determinant but lacks P Qn the alternating signs, i.e., for an n × n matrix A = (aij), perA = π i=1 ai,π(i), where the sum is over all permutations π. In this section, we will relate the permanent of a Hessenberg matrix and the Tribonacci numbers. Let us consider the n-square lower-Hessenberg matrix:   2 , if j = i, for i, j = 2, . . . , n − 1;  1 , if j = i + 1 and i = j = 1; (1.7) Hn =  −1 , if j = i − 3;  0 , otherwise, i.e.,  1 1   0 2 1     0 0 2 1     −1 0 0 2 1  Hn =   .  ......   . . . . .     −1 0 0 2 1  −1 0 0 2 Then we have the following theorem. Theorem 1.1. Let us consider the matrix deﬁned in (1.7). Then

(1.8) per Hn = Tn+1 , 74 DURMUS BOZKURT, CARLOS M. DA FONSECA AND FATIH YILMAZ where Tn is nth Tribonacci number. Proof. The result is clear for n = 1, 2, 3, 4. It is not hard to see that, for n > 4, we have per Hn = 2 per Hn−1 − per Hn−4 . This means that per Hn = per Hn−1 + 2 per Hn−2 − per Hn−4 − per Hn−5

= per Hn−1 + per Hn−2 + 2 per Hn−3 − per Hn−4 − per Hn−5 − per Hn−6

= per Hn−1 + per Hn−2 + per Hn−3 +

+ per Hn−4 − per Hn−5 − per Hn−6 − per Hn−7 . .

= per Hn−1 + per Hn−2 + per Hn−3 + per H4 − per H3 − per H2 − per H1

= per Hn−1 + per Hn−2 + per Hn−3 . Therefore, we get (1.8). This theorem can also be proved by using contraction method deﬁned by Brualdi and Gibson in [16]. Notice that according to [4, 6], we also have  1 −1   0 2 −1     0 0 2 −1     −1 0 0 2 −1  det   = Tn+1 .  ......   . . . . .     −1 0 0 2 −1  −1 0 0 2 n×n References [1] D. Bozkurt, T.-Y. Tam, Determinants and Inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers, Appl. Math. Comput. 219 (2012), no.2, 544-551. [2] A. Dasdemir, On the Pell, Pell-Lucas and modiﬁed Pell numbers by matrix method. Appl. Math. Sci. (Ruse) 5 (2011), no. 61-64, 3173-3181. [3] P.J. Davis, Circulant Matrices, Wiley, NewYork, 1979. [4] C.M. da Fonseca, An identity between the determinant and the permanent of Hessenberg-type matrices, Czechoslovak Math. J. 61(136) (2011), no.4, 917-921. [5] C.M. da Fonseca, On the location of the eigenvalues of Jacobi matrices, Appl. Math. Lett. 19 (2006), no.11, 1168-1174. [6] P.M. Gibson, An identity between permanents and determinants, Amer. Math. Monthly 76 (1969), 270-271. [7] H. Karner, J. Schneid, C.W. Ueberhuber, Spectral decomposition of real circulant matrices, Linear Algebra Appl. 367 (2003), 301-311. [8] T. Koshy, Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001. [9] G. Y. Lee, k-Lucas numbers and associated bipartite graphs, Linear Algebra Appl. 320 (2000), no.1-3, 51-61. [10] S.-Q. Shen, J.-M. Cen, Y. Hao, On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput. 217 (2011), no.23, 9790-9797. [11] N.J.A. Sloane, The on-line encyclopedia of integer sequences, 1999, http://oeis.org/A000073 [12] F. Yılmaz, D. Bozkurt, Hessenberg matrices and the Pell and Perrin numbers, J. Number Theory 131 (2011), no.8, 1390-1396. THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES 75

[13] F. Zhang, Matrix Theory, Basic Results and Techniques, Springer, New York, 2011. [14] G. Zhao, The improved nonsingularity on the r-circulant matrices in signal processing, In- ternational Conference on Computer Technology and Development - ICCTD 2009, Kota Kinabalu, 564-567. [15] E. Gokcen Alptekin, T. Mansour,and N. Tuglu, Norms of Circulant and Semicirculant matrices and Horadam’s sequence, Ars combinatorica 85 (2007) 353–359. [16] R.A. Brualdi, P.M. Gibson, Convex polyhedra of doubly stochastic matrices I: applications of the permanent function, J. Combin. Theory A 22 (1977), 194-230.

Department of Mathematics, Selcuk University, 42250 Konya, Turkey. E-mail address: [email protected]

Department of Mathematics, Kuwait University, Safat 13060, Kuwait. E-mail address: [email protected]

Department of Mathematics, Gazi University, 06900 Ankara, Turkey. E-mail address: [email protected]