^{1}

^{*}

^{1}

^{*}

^{1}

Let

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

where and are given matrices. For instance, Cvetković-Ilić [

We begin this section with the following lemma, which can be deduced from [

Lemma 1. (Theorem 3.1 in [

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

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

where. Partitioning

(3) is equivalent to

Therefore,

Partition and denote

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).

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).