Generalized Reflexive Matrices: Special Properties and Applications

The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem

“…We first assume that there is a Hermitian matrix A reflexive with respect to J such that AX = XD. By a similar reasoning as before, the form of the matrix A is given by (4). Substituting partition (6) in AX = XD, it is obtained that…”

“…For instance, the problem for Hermitian matrices anti-reflexive with respect to a generalized reflection (J 2 = I n and J * = J) was solved in [14]. The optimization problem related to reflexive matrices with respect to a pair of generalized reflections was studied in [4]. The inverse eigenvalue problem for Hermitian reflexive (antireflexive) matrices with respect to a Hermitian tripotent matrix was analyzed in [9].…”

On the Reflexive Solutions of the Matrix Equation Axb + Cyd = E

“…The following two special classes of subspaces in C n×n C n×n r (P ) = {A ∈ C n×n : A = P AP }, C n×n a (P ) = {A ∈ C n×n : A = −P AP }, are proposed by Chen [3], Chen and Sameh [5]. The matrices A in C n×n r (P ) and B in C n×n a (P ) are, respectively, said to be the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P .…”

“…The matrices A in C n×n r (P ) and B in C n×n a (P ) are, respectively, said to be the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P . The reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and scientific computations [3,4,5]. We know that matrix equation is one of the topics of very active research in the computational mathematics, and a large number of papers have presented various methods for solving several matrix equations [8,11,12,25,26,27,28].…”

The generalized anti-reflexive solutions for a class of matrix equations (BX = C, XD = E)

“…OC n×n denote the set of n × n unitary matrices. To extend reflexive (anti-reflexive) matrices and centrosymmetric matrices, Chen [11] has introduced two new special classes of matrices, which are generalized reflexive matrices and generalized anti-reflexive matrices. He presented three examples obtained from the altitude estimation of a level network, an electric network and structural analysis of trusses.…”