23 research outputs found
On Jacobsthal and Jacobsthal-Lucas Circulant Type Matrices
Circulant type matrices have become an important tool in solving fractional order differential equations.
In this paper, we consider the circulant and left circulant and g-circulant matrices with
the Jacobsthal and Jacobsthal-Lucas numbers. First, we discuss the invertibility of the circulant
matrix and present the determinant and the inverse matrix. Furthermore, the invertibility of the
left circulant and g-circulant matrices is also discussed. We obtain the determinants and the inverse
matrices of the left circulant and g-circulant matrices by utilizing the relation between left
circulant, g-circulant matrices, and circulant matrix, respectively
On the Explicit Formula for Eigenvalues, Determinant, and Inverse of Circulant Matrices
Determining eigenvalues, determinants, and inverse for a general matrix is computationally hard work, especially when the size of the matrix is large enough. But, if the matrix has a special type of entry, then there is an opportunity to make it much easier by giving its explicit formulation. In this article, we derive explicit formulas for determining eigenvalues, determinants, and inverses of circulant matrices with entries in the first row of those matrices in any formation of a sequence of numbers. The main method of our study is exploiting the circulant property of the matrix and associating it with cyclic group theory to get the results of the formulation. In every discussion of those concepts, we also present some computation remarks
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, 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
On the Mersenne sequence
From Binet’s formula of Mersenne sequence we get some properties for
this sequence. Mersenne, Jacobsthal and Jacobsthal-Lucas sequences are
considered in order to achieve some relations between them, sums and certain
products involving terms of these sequences. We also present some results
with matrices involving Mersenne numbers such as the generating matrix,
tridiagonal matrices and circulant type matrices