Abstract Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

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具有子矩阵约束的广义中心对称矩阵的左右逆特征对问题

摘要 左、右特征对反问题是一个特殊的特征值反问题。关于这个问题有很多有意义的结果。然而，很少有作者考虑过具有子矩阵约束的左右逆特征对问题。在本文中，我们将考虑广义中心对称矩阵的具有领先主子矩阵约束的左右逆特征对问题及其最优逼近问题。结合左右特征对的特殊性质和广义奇异值分解，我们推导出问题的可解性条件及其一般解。利用正交变换下Frobenius范数的不变性，我们得到了最优逼近问题的唯一解。我们提出了一种算法和数值实验来给出最佳近似解。我们的结果扩展并统一了左右逆特征对问题和具有子矩阵约束的中心对称矩阵的逆特征值问题的许多结果。