1. No category

Mathematical Sciences And Applications E-Notes
c
Volume 2 No. 2 pp. 67–75 (2014) 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 :=
numbers c0 , c1 , . . . , cn−1 , is defined as

c0
c1
 cn−1 c0


..
Cn =  ...
.

 c2
c3
c1
c2
circ (c0 , c1 , . . . , cn−1 ), associated with the
. . . cn−2
. . . cn−3
..
..
.
.
...
c0
. . . cn−1
cn−1
cn−2
..
.
c1
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]:
λj =
n−1
X
ck ω jk ,
j = 0, 1, . . . , n − 1 ,
k=0
where ω = exp( 2πi
n ) and i =
√
−1, and the corresponding eigenvectors are
xj = (1, ω j , ω 2j , . . . , ω (n−1)j )T ,
j = 0, 1, . . . , n − 1 .
Therefore, we can write determinant of a nonsingular circulant matrix as:
!
n−1
Y n−1
X
det Cn =
ck ω jk
j=0
where k = 0, 1, . . . , n − 1.
A skew circulant matrix with the
square matrix of the form

a0
a1
 −an−1 a0


..
..

.
.

 −a2
−a3
−a1
−a2
k=0
first row (a0 , a1 , ..., an−1 ) is meant to be a
...
...
..
.
an−2
an−3
..
.
an−1
an−2
..
.
...
...
a0
−an−1
a1
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 Jacobsthal 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 polynomials 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,
Un (x) =
sin(n + 1)θ
,
sin θ
with x = cos θ
(0 6 θ < π),
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


√ n
 c ... ...

a


√
det 
=
bc Un
.

.. ..
2 bc

.
. b 
c a n×n
Lemma 1.1. If

(1.1)




An = 




d1
a
c
d2
b
a
d3
c
a
..
.
···
dn−1
dn









b
..
.
..
.
c
..
.
a

b
then
(1.2)
det An =
n
X
k=1
n−k
dk b
√ k−1
a
√
− bc
Uk−1
,
2 bc
where Uk (x) is the kth Chebyshev polynomial of second kind.
Proof. It is clear that
(1.3)
det An = b det An−1 + (−)n−1 dn
√ n−1
a
√
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 


 0 tn−3 0 . . . 1 0 


n−4

0
0 
(1.4)
Qn =  0 t

 ..
..
..
. .. .. 
.
 .
.
. .
. . 


 0
t
1 ... 0 0 
0
1
0 ... 0 0
here t is the positive root of the characteristic equation xn t2 + yn t + zn = 0, i.e.,
p
−yn + yn2 − 4xn zn
t=
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

1
 −1

 −1

 −1

 0


Pn =  0
 0

 0


..

.

 0
0
n-square matrix Pn as below:

0
0
0
0
0
0
0
0
0
1 

0
0
0
1 −1 

0
0
1 −1 −1 

0
1 −1 −1 −1 

1 −1 −1 −1
0 

−1 −1 −1
0
0 

−1 −1
0
0
0 

..
0
0
0 

...
...
...
...
...
.
..
..
.. 
1 −1 −1 −1
0
0
.
.
. 
−1 −1 −1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
..
.
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
−1
It can be seen that for all n > 3,
det(Pn ) = det(Qn ) =
1, n ≡ 1 or 2 mod 4
−1, n ≡ 0 or 3 mod 4.
where Qn is defined in (1.4) and det(Pn Qn ) = 1. By matrix multiplication, we get;
Sn = Pn Tn Qn
(1.5)
i.e.,







Sn = 





1 fn
0 gn
0 hn
0 0
0 0
..
..
.
.
0 0
0 0
Tn−1
Tn − Tn−1
yn + 1
yn
zn
Tn−2
Tn−1 − Tn−2
Tn−3
xn
yn
..
.
Tn−3
Tn−2 − Tn−3
Tn−4
···
···
···
xn
..
.
..
zn
T3
T4 − T3
T2
T2
T3 − T2
T1
xn
yn
xn
.
yn
zn
where
fn
= −
n
X
zi tn−i ,
i=2
gn
=
n−1
X
(Ti+1 − Ti )tn−i + xn−1 ,
i=2
hn
= −
n−2
X
zi−1 tn−i + (yn + 1)t + zn .
i=2
A simple application of the Laplace expansion in (1.5), determinant will be
|Sn | = gn
hn
0
0
..
.
0
0
Tn − Tn−1
yn + 1
yn
zn
Tn−1 − Tn−2
Tn−3
xn
yn
..
.
Tn−2 − Tn−3
Tn−4
···
···
xn
..
.
zn
..
.
yn
zn
T4 − T3
T2
xn
yn
T3 − T2 T1
xn













THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES
Multiplying the first row with −m =
71
hn
and adding it to the second row in Sn ,
gn
we obtain
|Sn | = gn
0
0
0
Tn −T n−1
m(T n −T n−1 ) + y n +1
yn
zn
..
.
Tn−1 −T n−2
m(T n−1 −T n−2 ) + T n−3
xn
yn
xn
..
..
.
···
···
.
zn
0
0
..
T3 −T 2
m(T 3 −T 2 ) + T 1
.
yn
zn
xn
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
det Tn
= gn [m(Tn − Tn−1 ) + yn + 1]xn−1
+
n
n−2
X
√
yn
k−1
n−k
Uk−1
.
+ gn
[m(Tn−k+1 − Tn−k ) + Tn−k−1 ]xn (− xn zn )
√
2 xn zn
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 


 0 mn−3 0 . . . 1 0 


n−4

0
0 
Mn =  0 m

 ..
..
..
. .. .. 
.
 .
.
. .
. . 


 0
m
1 ... 0 0 
0
1
0 ... 0 0
here m is the positive root of the characteristic equation an m2 + bn m + cn = 0, i.e.,
p
−bn + b2n − 4an cn
m :=
2an
here an = 1 + Tn+1 , bn = (Tn + Tn−1 ) and cn = Tn .
Then consider n-square matrix Nn as below:


0
0
0
0
0
0
0
0
0
1 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
It can be seen that det(Mn ) = det(Nn ) =
n ≡ 1 or 2 mod 4
for n > 3.
n ≡ 0 or 3 mod 4.
1,
−1,
Using matrix multiplication, we get
Wn := Nn Kn Mn
i.e.,
(1.6) 
fn∗
gn∗
h∗n
0
0
..
.
1
0
0
0
0
..
.






Wn = 




 0
0
Tn
Tn−1 − Tn
bn + 1
bn
cn
Tn−1
Tn−2 − Tn−1
−Tn−3
an
bn
..
.
Tn−2
Tn−3 − Tn−2
−Tn−4
···
···
···
an
..
.
..
T2
T2 − T3
−T1
an
bn
an
T3 − T4
−T2
T2 − T3
−T1
an
bn
an
.
bn
cn
cn
0
0
T3
T3 − T4
−T2













here
fn∗
=
n
X
Ti mn−i ,
i=2
gn∗
=
n−1
X
(Ti − Ti+1 )mn−i + Tn + 1,
i=2
h∗n
= −
n−2
X
Ti−1 mn−i + (bn + 1)m + cn .
i=2
Using Laplace expansion in (1.6),
|Wn | = gn∗
h∗n
0
0
..
.
Tn−1 − Tn
bn + 1
bn
cn
Tn−2 − Tn−1
−Tn−3
an
bn
..
.
0
0
gn∗
0
0
..
.
0
0
···
···
an
..
.
..
cn
Multiplying the first row with s := −
|Wn | = Tn−3 − Tn−2
−Tn−4
.
bn
cn
h∗n
and adding it to the second row in Wn
gn∗
Tn−1 − Tn
s(Tn−1 − Tn ) + bn + 1
bn
Tn−2 − Tn−1
s(Tn−2 − Tn−1 ) − Tn−3
an
cn
bn
..
.
···
···
..
..
T3 − T4
s(T3 − T4 ) − T2
T2 − T3
s(T2 − T3 ) − T1
an
bn
an
.
.
cn
THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES
whose determinant is equal to
s(Tn−1 − Tn ) + bn + 1 s(Tn−2 − Tn−1 ) − Tn−3
bn
an
cn
bn
= gn∗ .
..
···
..
73
s(T3 − T4 ) − T2
.
..
.
cn
an
bn
s(T2 − T3 ) − T1 .
an
Using Lemma 1.1, we get
det Wn
= gn∗ [s(Tn−1 − Tn ) + bn + 1]bn−1
+
n
n−2
X
p
an
∗
n−k
k−1
√
Uk−1
.
+ gn
[s(Tn−k − Tn−k+1 ) − Tn−k−1 ]bn (− bn cn )
2 bn cn
k=2
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
P Qnbut lacks
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.,






Hn = 




1
0
0
−1
1
2
0
0
..
.

1
2
0
..
.
−1
1
2
..
.
0
−1
1
..
.
0
0
..
.
2
0





.



1 
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
= Tn+1 .
det 



.
.
.
.
.
..
..
..
..
..





−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, International 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]