Research On Spectral Norms Of Special Matrices And Related Problems

Special matrix is an important part of matrix theory,which have always been of interest and continuously studied by scholars.In this paper,based on some combination methods,the structure of spe-cial matrices and related properties,combining with the identities of polyno-mials and numbers,we mainly study the spectral norms of special matrices,some arithmetic properties of Chebyshev polynomials and Legendre polyno-mials.The main contents are as follows:1.We study a new better estimation of the upper and lower bounds of spectral norms of geometric circulant matrix and r-circulant matrices involv-ing generalized k-Horadam numbers,when the value is r=1,the spectral norms of circulant matrix with generalized k-Horadam numbers can be ob-tained.2.The spectral norms of r-circulant matrix and geometric circulant ma-trix involving trigonometric functions cos(k?/n)and sin(k?/n)are studied;then,we give the spectral norms of circulant matrices and alternately positive and negative entrywise r-circulant matrices with cos2(k?/n),sin2(k?/n);the spectral norms of r-circulant matrices with reciprocal of Chebyshev polynomials and trigonometric functions are given.3.By using the properties of trigonometric function and exponential function,we study the spectral norms of r-Toeplitz matrices and r-Hankel matrices involving trigonometric function cos(k?/n),sin(k?/n)and exponential function e(k/n).4.Based on methods mentioned above,the spectral norms of Hankel-Hessenberg and Toeplitz-Hessenberg matrices involving trigonometric func-tion cos(k?/n),sin(k?/n)and exponential function e(k/n)are obtained.5.The power sum formula of Chebyshev polynomials and some integral formula of Legendre polynomials are obtained.