Linear restriction problem of Hermitian reflexive matrices and its approximation
The following two problems are considered. Problem I: Find a matrix \(A \in H\mathbb C_r^{n \times n}(Q)\) such that \(AX=B\) holds, where \(X, B \in \mathbb C^{n \times m}\) are given matrices and \(H\mathbb C_r^{n \times n}(Q)\) denotes the set of all \(n \times n\) Hermitian reflexive matrices associated with the generalized reflection matrix \(Q\). Problem II: Find a matrix \(A^\ast \in S_A\) such that \(\| A^\ast - {\bar A} \| = \min_{A \in S_A} \| A - {\bar A} \| \), where \({\bar A} \in\mathbb C^{n \times n}\) is a given matrix and \(S_A\) is the solution set of Problem I. At first the structure and properties of the space \(H\mathbb C_r^{n \times n}\) and of reflexive as well as anti-reflexive vectors are introduced. Additionally, their relationship is discussed. Using these properties Problem I is converted to an equivalent problem. Then the solvability of Problem I is investigated. Furthermore, the existence and uniqueness of the solution of Problem II is proved. Finally, for solving Problem II an algorithm is formulated and demonstrated by two examples.
scientific article; zbMATH DE number 5289109
scientific article