We considered the symmetric orthogonal anti-symmetric solution and its optimal approximation problem of the inverse quadratic eigenvalue problem. The matrix QR decomposition is used to transform the quadratic inverse eigenvalue problem into an equivalent system of equations, and then the symmetric orthogonal anti-symmetric solution and the general solution expression in the case of solutions are obtained. The existence and uniqueness of the solution of the best approximation problem are proved, and the expression of the best approximation solution is obtained.