Determinants and inverses of r-circulant matrices associated with a number sequence
TL;DR: In this article, the -circulant matrix associated with the numbers defined by the recursive relation with initial conditions and, where and, are obtained some formulas for the determinants and inverses of.
Abstract: Let be the -circulant matrix associated with the numbers defined by the recursive relation with initial conditions and , where and We obtain some formulas for the determinants and inverses of . Some bounds for spectral norms of are obtained as applications.
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TL;DR: In this paper, the spectral norm of r-circulant matrices with Fibonacci and Lucas numbers is estimated for the first row of the first column of the matrix.
Abstract: Let us define $A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$
to be a $n\times n$
r-circulant matrix. The entries in the first row of $A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$
are $a_{i}=F_{i}$
, or $a_{i}=L_{i}$
, or $a_{i}=F_{i}L_{i}$
, or $a_{i}=F_{i}^{2}$
, or $a_{i}=L_{i}^{2}$
(
$i=0,1,\ldots,n-1$
), where $F_{i}$
and $L_{i}$
are the ith Fibonacci and Lucas numbers, respectively. This paper gives an upper bound estimation of the spectral norm for r-circulant matrices with Fibonacci and Lucas numbers. The result is more accurate than the corresponding results of S Solak and S Shen, and of J Cen, and the numerical examples have provided further proof.
25 citations
TL;DR: In this article, the Moore-Penrose inverse of a nonsingular k-circulant matrix with geometric sequence is derived. But the method is not suitable for nonzero complex numbers.
Abstract: Let k be a nonzero complex number. In this paper we show how the inverse of a nonsingular k-circulant matrix can be obtained. The method is used to determine the inverse of a nonsingular k-circulant matrix with geometric sequence. If k = 1, then we get the result presented in the paper A.C.F. Bueno, Right Circulant Matrices With Geometric Progression, Int. J. Appl. Math. Res. 1(4) (2012), 593– 603. Also, we derive the Moore-Penrose inverse of a singular k-circulant matrix with geometric sequence. At the end of the paper, we illustrate the obtained results by examples.
21 citations
TL;DR: Bahsi and Solak as discussed by the authors considered k-circulant matrices with arithmetic sequence and investigated the eigenvalues, determinants and Euclidean norms of such matrices.
Abstract: Let k be a nonzero complex number. In this paper we consider k - circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k=1 , the inverses of such ( invertible ) matrices are obtained (in a way different from the way presented in the paper: M. Bahsi and S. Solak , On the Circulant Matrices with Arithmetic Sequence , Int. J. Contemp . Math. Sci. 5 (25) (2010), 1213-1222, and the Moore-Penrose inverses of such (singular) matrices are derived.
19 citations
Cites background from "Determinants and inverses of r-circ..."
...The paper [5] is devoted to obtaining the determinants and inverses of k-circulant matrices associated with a number sequence....
[...]
01 Dec 2018
TL;DR: In this article, the authors studied properties of restricted and associated Fubini numbers and introduced modified Bernoulli and Cauchy numbers, and studied characteristic properties of these numbers.
Abstract: We study some properties of restricted and associated Fubini numbers. In particular, they have the natural extensions of the original Fubini numbers in the sense of determinants. We also introduce modified Bernoulli and Cauchy numbers and study characteristic properties.
14 citations
TL;DR: In this paper, the spectral norm of r-circulant matrices with the Pell and Lucas numbers is estimated for r-Circulant matrix with the first row entries in the first column.
Abstract: Let us define $A=C_{r}(a_{0},a_{1},\ldots,a_{n-1})$
to be a $n \times n$
r-circulant matrix. The entries in the first row of $A=C_{r}(a_{0},a_{1},\ldots,a_{n-1})$
are $a_{i}=P_{i}$
, $a_{i}=Q_{i}$
, $a_{i}=P_{i}^{2}$
or $a_{i}=Q_{i}^{2}$
(
$i=0, 1, 2, \ldots, n-1$
), where $P_{i}$
and $Q_{i}$
are the ith Pell and Pell-Lucas numbers, respectively. We find some bounds estimation of the spectral norm for r-Circulant matrices with Pell and Pell-Lucas numbers.
11 citations
References
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01 Jan 2001
TL;DR: The first 100 Lucas Numbers and their prime factorizations were given in this article, where they were shown to be a special case of the first 100 Fibonacci Numbers and Lucas Polynomials.
Abstract: Preface. List of Symbols. Leonardo Fibonacci. The Rabbit Problem. Fibonacci Numbers in Nature. Fibonacci Numbers: Additional Occurrances. Fibonacci and Lucas Identities. Geometric Paradoxes. Generalized Fibonacci Numbers. Additional Fibonacci and Lucas Formulas. The Euclidean Algorithm. Solving Recurrence Relations. Completeness Theorems. Pascal's Triangle. Pascal-Like Triangles. Additional Pascal-Like Triangles. Hosoya's Triangle. Divisibility Properties. Generalized Fibonacci Numbers Revisited. Generating Functions. Generating Functions Revisited. The Golden Ratio. The Golden Ratio Revisited. Golden Triangles. Golden Rectangles. Fibonacci Geometry. Regular Pentagons. The Golden Ellipse and Hyperbola. Continued Fractions. Weighted Fibonacci and Lucas Sums. Weighted Fibonacci and Lucas Sums Revisited. The Knapsack Problem. Fibonacci Magic Squares. Fibonacci Matrices. Fibonacci Determinants. Fibonacci and Lucas Congruences. Fibonacci and Lucas Periodicity. Fibonacci and Lucas Series. Fibonacci Polynomials. Lucas Polynomials. Jacobsthal Polynomials. Zeros of Fibonacci and Lucas Polynomials. Morgan-Voyce Polynomials. Fibonometry. Fibonacci and Lucas Subscripts. Gaussian Fibonacci and Lucas Numbers. Analytic Extensions. Tribonacci Numbers. Tribonacci Polynomials. Appendix 1: Fundamentals. Appendix 2: The First 100 Fibonacci and Lucas Numbers. Appendix 3: The First 100 Fibonacci Numbers and Their Prime Factorizations. Appendix 4: The First 100 Lucas Numbers and Their Prime Factorizations. References. Solutions to Odd-Numbered Exercises. Index.
1,250 citations
TL;DR: In this paper, Popescu et al. discuss necessary and sufficient conditions for circulant matrices to be non-singular, and various distinct representations they have, goals that allow us to lay out the rich mathematical structure that surrounds them.
Abstract: Some mathematical topics, circulant matrices, in particular, are pure gems that cry out to be admired and studied with different techniques or perspectives in mind. Our work on this subject was originally motivated by the apparent need of one of the authors (IK) to derive a specific result, in the spirit of Proposition 24, to be applied in his investigation of theta constant identities [9]. Although progress on that front eliminated the need for such a theorem, the search for it continued and was stimulated by enlightening conversations with Yum-Tong Siu during a visit to Vietnam. Upon IK’s return to the US, a visit by Paul Fuhrmann brought to his attention a vast literature on the subject, including the monograph [4]. Conversations in the Stony Brook Mathematics’ common room attracted the attention of the other author, and that of Sorin Popescu and Daryl Geller∗ to the subject, and made it apparent that circulant matrices are worth studying in their own right, in part because of the rich literature on the subject connecting it to diverse parts of mathematics. These productive interchanges between the participants resulted in [5], the basis for this article. After that version of the paper lay dormant for a number of years, the authors’ interest was rekindled by the casual discovery by SRS that these matrices are connected with algebraic geometry over the mythical field of one element. Circulant matrices are prevalent in many parts of mathematics (see, for example, [8]). We point the reader to the elegant treatment given in [4, §5.2], and to the monograph [1] devoted to the subject. These matrices appear naturally in areas of mathematics where the roots of unity play a role, and some of the reasons for this to be so will unfurl in our presentation. However ubiquitous they are, many facts about these matrices can be proven using only basic linear algebra. This makes the area quite accessible to undergraduates looking for research problems, or mathematics teachers searching for topics of unique interest to present to their students. We concentrate on the discussion of necessary and sufficient conditions for circulant matrices to be non-singular, and on various distinct representations they have, goals that allow us to lay out the rich mathematical structure that surrounds them. Our treatement though is by no means exhaustive. We expand on their connection to the algebraic geometry over a field with one element, to normal curves, and to Toeplitz’s operators. The latter material illustrates the strong presence these matrices have in various parts of modern and classical mathematics. Additional connections to other mathematics may be found in [11]. The paper is organized as follows. In §2 we introduce the basic definitions, and present two models of the space of circulant matrices, including that as a
792 citations
19 Sep 2001
435 citations
TL;DR: It is proved that A n is invertible for n > 2, and B n isInverted for any positive integer n .
80 citations
"Determinants and inverses of r-circ..." refers background in this paper
...In [7], the determinants and inverses of the circulant matrices An = circ (F1, F2, ....
[...]
TL;DR: In this article, spectral decompositions of real circulant matrices are presented, i.e., eigendecompositions and singular value decomposition, for skew right and skew left circulants.
76 citations