Left Circulant Research Articles

On the sum of left and right circulant matrices

We consider square matrices arising as the sum of left and right circulant matrices and derive asymptotics of the sequence of their powers. Particular emphasis is laid on the case where the matrix has consecutive integer entries; we find explicit formulae for the eigenvalues and eigenvectors of the matrix in this case and find its Moore-Penrose pseudoinverse. The calculation involves the discrete Fourier transform of integer vectors arising from sum systems and exhibits a resonance phenomenon.

New unitary equivalences for some operator matrices with applications

In this paper, first we consider cross-diagonal, in particular off-diagonal, operator matrices; those operator matrices such that all entries except those on the main and off diagonals are zero. We show that this operator matrix is unitarily equivalent to a block diagonal operator matrix whose diagonal blocks are all two-by-two, except at most one of them which is one-by-one. Then, using this unitary equivalence, we show that any left circulant operator matrix is unitarily equivalent to a direct sum of a one-by-one operator and an off-diagonal operator matrix. As an application, we give equalities for the numerical radius of some important operator matrices. In particular, for the following left circulant operator matrix, we show that where is the nth root of unity. Meanwhile, some inequalities for general operator matrices are obtained.

Explicit inverses of generalized Tribonacci circulant type matrices

In this paper, we consider generalized Tribonacci circulant type matrices, including the circulant and left circulant. Firstly, we discuss the invertibility of generalized Tribonacci circulant matrix and give the explicit determinant and inverse matrix based on constructing the transformation matrices. Afterwards, by utilizing the relation between circulant and left circulant, the invertibility of generalized Tribonacci left circulant matrix is also discussed. The determinant and inverse of generalized Tribonacci left circulant matrix are given respectively.

Determinants of binomial-related circulant matrices

Abstract Due to their rich algebraic structures and wide applications, circulant matrices have been of interest and continuously studied. In this paper, n×n complex left and right circulant matrices whose first row consists of the coefficients in the expansion of (x + zy)n−1 are focused on, where z is a nonzero complex number and n is a positive integer. In the case where z ∈ {1, −1, i, −i}, explicit formulas for the determinants of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as the special case where z = 1. Finally, some remarks and open problems are discussed.

A note on certain matrices with h(x) – Fibonacci quaternion polynomials

In this paper we consider a g – circulant, right circulant, left circulant and a special kind of a tridiagonal matrices whose entries are h(x) – Fibonacci quaternion polynomials. We present the determinant of these matrices and with the tridiagonal matrices we show that the determinant is equal to the nth term of the h(x) – Fibonacci quaternion polynomial sequences.

Exact determinants and inverses of generalized Lucas skew circulant type matrices

In this paper, we consider generalized Lucas skew circulant type matrices, including the skew circulant and skew left circulant. Firstly, we discuss the invertibility of generalized Lucas skew circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of generalized Lucas skew left circulant matrix is also discussed. The determinant and the inverse matrix of generalized Lucas skew left circulant matrix are obtained respectively.

Explicit inverse matrices of Tribonacci skew circulant type matrices

The determinants and inverses of Tribonacci skew circulant type matrices are discussed in the paper. Firstly, Tribonacci skew circulant type matrices are defined. In addition, we show the invertibility of the Tribonacci skew circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between circulant and left circulant, the invertibility of the Tribonacci skew left circulant are also discussed. Finally, the determinants and the inverse matrices of these matrices are given, respectively.

VanderLaan Circulant Type Matrices

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 VanderLaang-circulant matrix.

Explicit Form of the Inverse Matrices of Tribonacci Circulant Type Matrices

It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant,g-circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonaccig-circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

Circulant Type Matrices with the Sum and Product of Fibonacci and Lucas Numbers

Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant andg-circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the left circulant andg-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant andg-circulant matrices by utilizing the relation between left circulant, andg-circulant matrices and circulant matrix, respectively.

Determinants of the RFMLR Circulant Matrices with Perrin, Padovan, Tribonacci, and the Generalized Lucas Numbers

The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences.

Algorithms for Finding Inverse of Two Patterned Matrices overZp

Circulant matrix families have become an important tool in network engineering. In this paper, two new patterned matrices overZpwhich include row skew first-plus-last right circulant matrix and row first-plus-last left circulant matrix are presented. Their basic properties are discussed. Based on Newton-Hensel lifting and Chinese remaindering, two different algorithms are obtained. Moreover, the cost in terms of bit operations for each algorithm is given.

Determinants, Norms, and the Spread of Circulant Matrices with Tribonacci and Generalized Lucas Numbers

Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degreen, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

Guo perturbation for symmetric nonnegative circulant matrices

Guo[W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261–270] sets the question: if the list Λ = { λ 1 , λ 2 , … , λ n } is symmetrically realizable (that is, Λ is the spectrum of a symmetric nonnegative matrix), and t > 0 , whether or not the list Λ t = { λ 1 + t , λ 2 ± t , λ 3 , … , λ n } is also symmetrically realizable. In this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant. We also give a necessary and sufficient condition for Λ to be the spectrum of a nonnegative left circulant matrix.