In this talk, we consider the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by Chen and Chu [SIAM J. Numer. Anal., 33 (1996), pp. 2417–2430]. We provide a geometric Gauss-Newton method for solving the least squares inverse eigenvalue problem. The global and local convergence analysis of the proposed method is established under some assumptions. Also, a preconditioned conjugate gradient method with an efficient preconditioner is proposed for solving the geometric Gauss–Newton equation. Finally, some numerical tests, including an application in the inverse Sturm-Liouville problem, are reported to illustrate the efficiency of the proposed method.