对称正交对称半正定矩阵逆特征值问题

Inverse Eigenvalue Problem of Symmetric Ortho-Symmetric Positive Semi-Definite Matrices

对给定的特征值和对应的特征向量,提出了对称正交对称半正定矩阵逆特征值问题及最佳逼近问题.通过分析对称正交矩阵和对称正交对称半正定矩阵的结构,利用矩阵的奇异值分解,导出了这种逆特征值问题的最小二乘解的表达式,以及这种逆特征值问题相容的充要条件和通解表达式.利用矩阵的极分解,导出了逆特征值问题的最佳逼近解.最后,通过数值算例说明了如何计算矩阵逆特征值问题的最小二乘解及最佳逼近解.

From given eigenvalues and eigenvectors, the inverse eigenvalue problem of symmetric ortho-symmetric positive semi-definite matrices and its optimal approximate problem were considered. By analyzing the structure of symmetric orthogonal matrices and symmetric ortho-symmetric positive semi-definite matrices and by applying the singular value decomposition of matrices, the expression of the least-squares solutions of this inverse eigenvalue problem was derived. Moreover,the sufficient and necessary conditions for the consistency of this inverse eigenvalue problem and the expression of the solutions also were given. The optimal approximate solution of this inverse eigenvalue problem also was given by means of the polar decomposition of matrices. In the end, a numerical example was given to show how to compute the least-squares solutions and the optimal approximate solution.