The limit of the m-norms of a class of symmetric matrices and its applications

We consider a special symmetric matrix and obtain a similar formula as the one obtained by Weyl’s criterion. Some applications of the formula are given, where we give a new way to calculate the integral of $\ln \mathrm{\Gamma }(x)$ on $[0,1]$, and we claim that one class of matrices are not Hadamard matrices.