On the basis of conjugate gradient method

an iterative method was presented to solve the least squares symmetric solution of Lyapunov matrix equation.By the iterative method

the solvability of the equation over symmetric solution can be determined.Whether the matrix equation is consistent or not

the least squares symmetric solution can be obtained automatically within finite iteration steps.And the symmetric solution with least norm can be obtained by choosing a special initial symmetric matrix.In addition

its optimal approximation matrix to a given matrix can be obtained.The given numerical examples show that the iterative method is quite efficient.