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In this paper, the left and right inverse eigenpairs problem of orthogonal matrices and its optimal approximation solution are considered. Based on the special properties of eigenvalue and the special relations of left and right eigenpairs for orthogonal matrices, we find the equivalent problem, and derive the necessary and sufficient conditions for the solvability of the problem and its general solutions. With the properties of continuous function in bounded closed set, the optimal approximate solution is obtained. In addition, an algorithm to obtain the optimal approximation and numerical example are provided

Left and Right Inverse Eigenpairs Problem of Orthogonal Matrices