1. Introduction
Throughout we denote the complex 
matrix space by
. The symbols 
and 
stand for the identity matrix with the appropriate size, the conjugate transpose, the inverse, and the Frobenius norm of
, respectively.
The reflexive matrices have extensive applications in engineering and scientific computation. It is a very active research topic to investigate the reflexive solution to the linear matrix equation
(1)
where 
and 
are given matrices. For instance, Cvetković-Ilić [1] and Peng et al. [2] have given the necessary and sufficient conditions for the existence and the expressions of the reflexive solutions for the matrix Equation (1) by using the structure properties of matrices in required subset of 
and the generalized singular value decomposition (GSVD); Different from [1,2], Ref. [3] has considered generalized reflexive solutions of the matrix Equation (1); in addition, Herrero and Thome [4] have found the reflexive (with respect to a generalized
—reflection matrix
) solutions of the matrix Equation (1) by the (GSVD) and the lifting technique combined with the Kronecker product.
2. The Reflexive Least Squares Solutions to Matrix Equation (1)
We begin this section with the following lemma, which can be deduced from [5].
Lemma 1. (Theorem 3.1 in [5]) Let the canonical correlation decomposition of matrix pair 
and 
with
. rank
, rank
, rank
, rank
be given as
where
with the same row partitioning, and
,
with the same row partitioning, and 
and let
Then general forms of least squares solutions 
of matrix equation
are as follows:
where
and 
are arbitrary matrices.
Theorem 2. Given 
. Then the reflexive least squares solutions to the matrix Equation (1) can be expressed as
(2)
where
and 
are arbitrary matrices.
Proof. It is required to transform the constrained problem to unconstrained ones. To this end, let
be the eigenvalue decomposition of the Hermitian matrix 
with unitary matrix
. Obviously, 
holds if and only if
(3)
where
. Partitioning
(3) is equivalent to
Therefore,
(4)
Partition 
and denote
(5)
According to (4), (5) and the unitary invariance of Frobenius norm
Applying Lemma 2.1, we derive the reflexive least squares solutions to matrix Equation (1) with the constraint 
which can be expressed as (2).
3. Acknowledgements
This research was supported by the Natural Science Foundation of Hebei province (A2012403013), the Natural Science Foundation of Hebei province (A2012205028) and the Education Department Foundation of Hebei province (Z2013110).