Abstract: We considered the generalized centrosymmetric solution (generalized anticentrosymmetric solution) of an inverse quadratic eigenvalue problem and its optimal approximation problem. By using the orthogonal projection methods of matrix, we gave the solution of matrix equation AX+BY+CZ=0and its optimal approximation problem. According to the properties of generalized centrosymmetric matrices (generalized anti-centrosymmetric matrices), we derived the conditions for the problem with a generalized centrosymmetric solution (generalized anticentrosymmetric solution) and the expression of general solution. We proved the existence and the uniqueness of solution of the optimal approximation problem, and obtained the expression of the optimal approximation solution.

Key words: optimal approximation solution, orthogonal projection method, generalized centrosymmetric matrix, inverse quadratic eigenvalue problem