R-circulant Matrix Research Articles

On the spectral norms of some circulant matrices with the trigonometric functions

In this paper, we use the properties of an r-circulant matrix and a geometric circulant matrix to study the spectral norms of the r-circulant matrix and the geometric circulant matrix involving trigonometric functions by some algebra methods.

The spectral norms of geometric circulant matrices with the generalized k-Horadam numbers

In this paper, we use the algebra methods, the properties of the r-circulant matrix and the geometric circulant matrix to study the upper and lower bound estimate problems for the spectral norms of a geometric circulant matrix involving the generalized k-Horadam numbers, and we obtain some sharp estimations for them. We can also give a new estimation for the norms of a r-circulant matrix involving the generalized k-Horadam numbers.

On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix

Abstract Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.

Explicit form of determinants and inverse matrices of Tribonacci r-circulant type matrices

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

On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers

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.

The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers

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.

The Discriminance for <i>FLDcirc<sub>r</sub></i> Matrices and the Fast Algorithm of Their Inverse and Generalized Inverse

This paper presents a new type of circulant matrices. We call it the first and the last difference r-circulant matrix (FLDcircr matrix). We can verify that the linear operation, the matrix product and the inverse matrix of this type of matrices are still FLDcircr matrices. By constructing the basic FLDcircr matrix, we give the discriminance for FLDcircr matrices and the fast algorithm of the inverse and generalized inverse of the FLDcircr matrices.

On the Spectrum and Spectral Norms of r-Circulant Matrices with Generalized k-Horadam Numbers Entries

This work is concerned with the spectrum and spectral norms of r-circulant matrices with generalized k-Horadam numbers entries. By using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them. In addition, a sufficient condition for an r-circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral norms of normal r-circulant matrix with generalized k-Horadam numbers, which generalize and improve the known results.

On the norms of an r-circulant matrix with the generalized k-Horadam numbers

In this paper, we present new upper and lower bounds for the spectral norm of an r-circulant matrix H = Cr(Hk,0, Hk,1, Hk,2, ... , Hk,n-1) whose entries are the generalized k-Horadam numbers. Furthermore, we obtain new formulas to calculate the eigenvalues and determinant of the matrix H. MSC: 11B39; 15A60; 15A15

On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices

Let n be a positive integer, and C n (r) the set of all n × n r-circulant matrices over the Boolean algebra B = {0, 1}, $$G_n = \bigcup\limits_{r = 0}^{n - 1} {C_n (r)} $$ . For any fixed r-circulant matrix C (C ≠ 0) in G n , we define an operation “*” in G n as follows: A * B = ACB for any A, B in G n , where ACB is the usual product of Boolean matrices. Then (G n , *) is a semigroup. We denote this semigroup by G n (C) and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix C. Let F be an idempotent element in G n (C) and M(F) the maximal subgroup in G n (C) containing the idempotent element F. In this paper, the elements in M(F) are characterized and an algorithm to determine all the elements in M(F) is given.

The pseudoinverse of an 𝑟-circulant matrix

It is shown that the Moore-Penrose pseudoinverse C + {C^ + } of an r-circulant matrix C is always the conjugate transpose of an r-circulant matrix. In addition, necessary and sufficient conditions are given for C + {C^ + } to be an s-circulant matrix. Finally, a method for calculating C + {C^ + } is given.