The Solutions to Matrix Equation AX = B with Some Constraints

with different constraints such as symmetric, reflexive, Hermitian-generalized Hamiltonian, and repositive definite [5–9]. By special matrix decompositions such as singular value decompositions (SVDs) and CS decompositions [10– 12], Hu and his collaborators [13–15], Dai [16], and Don [17] have presented the existence conditions and detailed representations of constrained solutions for (1) with corresponding constraints, respectively. For instance, Peng and Hu [18] presented the eigenvectors-involved solutions to (1) with reflexive and antireflexive constraints; Wang and Yu [19] derived the bi(skew-)symmetric solutions and the bi(skew-)symmetric least squares solutions with the minimum norm to this matrix equation; Qiu and Wang [20] proposed an eigenvectors-free method to (1) with PX = XP and X = sX constraints, where P is a Hermitian involutory matrix and s = ±1. Inspired by the work mentioned above, we focus on the matrix equation (1) with PX = XP and X = X constraints, which can be described as follows: findX such that


Introduction
Throughout we denote the complex  ×  matrix space by C × .The symbols ,  * ,  −1 , and ‖‖ stand for the identity matrix with the appropriate size, the conjugate transpose, the inverse, and the Frobenius norm of  ∈ C × , respectively.
Inspired by the work mentioned above, we focus on the matrix equation (1) with  =  and  * =  constraints, which can be described as follows: find  such that {‖ − ‖ 2 = min,  = ,  * = } . ( Moreover, we also discuss the least squares solutions of (1) with  =  * and  * =  constraints, where  is a given unitary matrix of order .
In Section 2, we present the least squares solutions to the matrix equation (1) with the constraints  =  and  * = .In Section 3, we derive the least squares solutions to the matrix equation (1) with the constraints  =  * and  * = .In Section 4, we give an algorithm and a numerical example to illustrate our results.

Least Squares Solutions to the
Matrix Equation (1) with the Constraints  =  and  * = It is required to transform the constrained problem to unconstrained one.To this end, let be the eigenvalue decomposition of the Hermitian matrix  with unitary matrix .Obviously,  =  holds if and only if diag where  =  * .
(4) is equivalent to Therefore, The constraint  * =  is equivalent to with Partition  = ( 1 ,  2 ) and denote then assume that the singular value decomposition of  1 and  2 is as follows: where  1 , where  14 =  * 14 and  24 =  * 24 are arbitrary matrix.
Proof.According to (8) and the unitary invariance of Frobenius norm By ( 9), the least squares problem is equivalent to We get According to (10), the least squares problem is equivalent to Assume that Then we have ) ) ) where  14 =  * 14 and  24 =  * 24 are arbitrary matrix.