Abstract
For a given generalized reflection matrix J, that is, JH=J, J2=I, where JH is the conjugate transpose matrix of J, a matrix A∈Cn×n is called a Hermitian (anti)reflexive matrix with respect to J if AH=A and A=±JAJ. By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB=C over complex field.
Highlights
Throughout this paper, we denote the set of all m × n matrices over the real field and complex field by R푚×푛 and C푚×푛, respectively
The set of all n × n Hermitian reflexive matrices with respect to J is denoted by HC푟푛×푛(J)
It follows from Lemma 6 that vec (g (EY퐻F)) = f (F퐻 ⊗ E) vec (g (Y퐻)), (37)
Summary
Throughout this paper, we denote the set of all m × n matrices over the real field and complex field by R푚×푛 and C푚×푛, respectively. We consider the Hermitian (anti)reflexive solution problem of matrix equation AXB = C. Given a generalized reflection matrix J, that is, J퐻 = J, J2 = I, a matrix A ∈ C푛×푛 is called a Hermitian reflexive matrix with respect to J if A퐻 = A and A = JAJ. The set of all n × n Hermitian reflexive matrices with respect to J is denoted by HC푟푛×푛(J). Finding the least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB = C is still a problem. The structure of Hermitian (anti)reflexive matrix promises the following decompositions, which would simplify our problems.
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