Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

• Main Author: Li Fan-Liang
• Format: Article
• Language: English
• Published: De Gruyter 2020-06-01
• Series: Open Mathematics
• Subjects: leading principal submatrix; submatrix constraint; generalized centrosymmetric matrix; left and right inverse eigenpairs; optimal approximation; 65f18; 15a24
• Online Access: https://doi.org/10.1515/math-2020-0020

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.