Left and right inverse eigenvalue problem of (R, S)-symmetric matrices and its optimal approximation problem

The left and right inverse eigenvalue problem, which mainly arises in perturbation analysis of matrix eigenvalue and recursive matters, has some practical applications in engineer and scientific computation fields. In this paper, we give the solvability conditions of and the general expressions to the left and right inverse eigenvalue problem for the (R, S)-symmetric and (R, S)-skew symmetric solutions. The corresponding best approximation problems for the left and right inverse eigenvalue problem are also solved. That is, given an arbitrary complex n-by-n matrix A∼, find a (R, S)-symmetric (or (R, S)-skew symmetric) matrix AA∼ which is the solution to the left and right inverse eigenvalue problem such that the distance between A∼ and AA∼ is minimized in the Frobenius norm. We give an explicit solution to the best approximation problem in the (R, S)-symmetric and (R, S)-skew symmetric solution sets of the left and right inverse eigenvalue problem under the assumption that R=R∗ and S=S∗. A numerical example is given to illustrate the effectiveness of our method.