On The Generalization Of Circulant Matrices And Their Application

Circulant matrices are one class of important special matrices, and have various generalizati-ons, and become active research fields of matrix theory increasingly.We study some problems on norms、determinants and inverses of some special matriceswith some combinatial numbers, disscuss fast algorithm for square root and spectral decompo-sition of one class of special matrices, the detail is as follows:1. By norm theory in matrix, we give upper and lower bounds for the spectral norms ofdiagonal factor circulant matrices with Fibonacci and Lucas numbers, then, we obtain somebounds for the spectral norms of Kronecker and Hadamard products of above matrices.2. By inequality between Euclidean norm and spectral norm、Binet formulas for generaliz-ed Fibonacci and Lucas numbers and scalar-valued polynomial for permutation factor circulantmatrices, we give upper and lower bounds for the spectral norms of permutation factor circulantmatrices with generalized Fibonacci and Lucas numbers.3. By inequality between Euclidean norm and spectral norm and scalar-valued polynomialfor Toeplitz matrices, we give upper and lower bounds for the spectral norms of Toeplitz matriceswith k Fibonacci and k Lucas numbers.4. By recursive relation of k Fibonacci and k Lucas numbers, we give expressions ofdeterminats of r circulant matrices A Cr(F_k,1,F_k,2,,F_k,n)and B C_r(L_k,1,L_k,2,,L_k,n),disscuss the invertible conditions for matrices A and B, also, we give expressions ofA1andB¹.5. By fast Fourier transform, we study fast algorithm for square root of permutation factorcirculant matrices, also, we disscuss spectral decomposition of above matrices.