Some Problems Of R-circulant Matrices With Some Combinatorial Numbers

r-Circulant matrices are an important component of the matrix theory and apply in many areas of molecular vibration, linear prediction, signal processing, coding theory, mathematical statistics etc. In recent years, the inverses and determinants of r-circulant matrices have been extensiveiy investigated, but there is no good result about the norms of some special r-circulant matrices. To this end, the norms of r-circulant matrices with some combinatorial numbers are studied in this article. In addition, although some algorithms for the inverses and determinants of r-circulant matrices are given by some scholars, the computational complexity of these algo-rithms are very amazing with the order of matrix increasing. The better results for the inverses and determinants of r-circulant matrices are obtained by the perfect properties of Fibonacci and Lucas numbers in this paper. The main contents of this article are organized as follows:In chapter 1, we introduce the history of k-Fibonacci and k-Lucas numbers, as well as the norms, determinants and inverses of some special matrices. Besides, we give some basic concepts and knowledge related to the norms, Kronecker and Hadamard products of the matrix.In chapter 2, we give the upper and lower bounds for the spectral norms of r-circulant matrices A= Cr(Cn0,1/2C1/n,…,1/nCnn-1) and B=Cr(0, Cn1,…,(n-1)Cnn-1) by some properties of Gamma function, where Cnk is the binomial coefficient. Afterwards, we obtain some bounds for the spectral norms of Hadamard and Kronecker products of these matrices.In chapter 3, we establish the upper and lower bounds for the spectral norms of r-circulant matrices A= Cr(Fk,0,Fk,1,…,Fk,n-1) andβ= Cr(Lk,0, Lk,1,…, Lk,n-1) by some properties of r-circulant matrices, where Fk,n and Lk,n are k-Fibonacci and k-Lucas numbers respectively. Afterwards, we obtain some bounds for the spectral norms of Hadamard and Kronecker products of these matrices.In chapter 4, by some properties of Fibonacci and Lucas numbers, we give the values of the determinants of r-circulant matrices An= Cr(F1, F2,…, Fn) and Bn= Cr(L1, L2,…, Ln). In addition, we discuss invertible conditions for matrices An and Bn, then obtain two explicit formulas for An-1 and Bn-1, and these formulas are only related to parameterγ, the Fibonacci and Lucas numbers. In the caseγ= 0, we also obtain respectively the inverses of the Fibonacci and the Lucas matrices.