A class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices. (English) Zbl 1080.15013

The paper deals with a class of constrained inverse eigenproblems and associated approximation problems.
A real matrix \(A\), of size \(n \times n\), is called skew symmetric and centrosymmetric if \(A^T=-A\) and \(JAJ=A\), where \(J\) denotes the \(n \times n\) “backward identity” matrix. The set of these matrices is denoted by \(ACSR^{n \times n}\).
The authors study the following problems:
Problem I. Let \(b>0\) a real number, \(Z\) an \(n \times k\), \(k<n\), complex matrix of rank \(k\) and \(D_0=\text{diag}(\lambda_1i,\ldots,\lambda_ki)\) with \(\lambda_j\) a real number, \(j=1,2,\ldots,k\). Find \((Z,D_0)\) such that the set \(\varphi(Z,D_0)= \{A \in ACSR^{n \times n} \mid AZ=ZD_0\}\) is nonempty and find the subset \(\varphi(Z,D_0,b) \subset \varphi(Z,D_0)\) such that the imaginary parts of all of the remaining eigenvalues of any matrix in \(\varphi(Z,D_0,b)\) are located in the interval \([-b,b]\).
Problem II. Given and \(n \times n\) real matrix \(B\), find \(A_B \in \varphi(Z,D_0,b)\) such that \[ \| B-A_B\| = \min_{A \in \varphi(Z,D_0,b)} \| B-A\| , \] where \(\| \cdot \| \) is the Frobenius norm.
The authors show that the mentioned problems are essentially decomposed into the same kind subproblems for real antisymmetric matrices with smaller dimensions. They present the general solution of Problems I and II in the real number field. They also obtain the explicit expression of the nearest matrix to a given matrix in the Frobenius norm.


15A18 Eigenvalues, singular values, and eigenvectors
15A29 Inverse problems in linear algebra
15B57 Hermitian, skew-Hermitian, and related matrices
Full Text: DOI


[1] Andrew, A. L., Eigenvectors of certain matrices, Linear Algebra Appl., 7, 151-162 (1973) · Zbl 0255.65021
[2] Trench, W. F., Spectral evolution of a one-parameter extension of a real symmetric Toeplitz matrix, SIAM J. Matrix Anal. Appl., 11, 601-611 (1990) · Zbl 0717.15008
[3] Trench, W. F., Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear Algebra Appl., 377, 207-218 (2004) · Zbl 1046.15028
[4] Trench, W. F., Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry, Linear Algebra Appl., 380, 199-211 (2004) · Zbl 1087.15013
[5] Xie, X. D.; Hu, X. Y.; Zhang, L., The solvability conditions for inverse eigenvalue problem of anti-bisymmetric matrices, J. Comput. Math., 20, 245-256 (2002) · Zbl 1005.65036
[6] Zhang, L., A class of inverse eigenvalue problems of symmetric matrices, Numer. Math. J. Chinese Univ., 12, 1, 65-71 (1990) · Zbl 0725.15007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.