Maximum rank decomposition and algorithm for unitary symmetric matrix

We study the algorithms for the unitary symmetric matrix, and prove the quantitative correspondence of F~*and G~*matrices of unitary symmetric matrices and its mother matrix A.And address two new algorithms for the unitary symmetric matrix at the same time.At last also analyse the Moore-Penrose inverse,e.g.g inverse, reflected ginverse,least square g inverse,least norm g inverse,and give the algorithms of Moore-Penrose inverse for unitary symmetric matrixtoo.Some examples illustrate the applications of the above theory.We can save dramatically the CPU time and memory.