1. Introduction
Throughout we denote the complex 
matrix space by 
the real 
matrix space by 
The symbols 
and 
stand for the identity matrix with the appropriate size, the conjugate transpose, the range, the null space, and the Frobenius norm of 
respectively. The Moore-Penrose inverse of 
denoted by 
is defined to be the unique matrix 
of the following matrix equations
Recall that an 
complex matrix 
is called 
(or range Hermitian) if 
matrices were introduced by Schwerdtfeger in [1], ever since many authors have studied 
matrices with entries from complex number field to semigroups with involution and given various equivalent conditions and many characterizations for matrix to be 
(see, [2-5]).
Investigating the matrix equation
(1)
with the unknown matrix 
being symmetric, reflexive, Hermitian-generalized Hamiltonian and re-positive definite is a very active research topic (see, [6-9]). As a generalization of (1), the classical system of matrix equations
(2)
has attracted many people’s attention and many results have been obtained about system (2) with various constraints, such as bisymmetric, Hermitian, positive semidefinite, reflexive, and generalized reflexive solutions, and so on (see, [9-12]). It is well-known that 
matrices are a wide class of objects that include many matrices as their special cases, such as Hermitian and skewHermitian matrices (i.e.,
), normal matrices (i.e.,
), as well as all nonsingular matrices. Therefore investigating the 
solution of the matrix Equation (2) is very meaningful.
Pearl showed in ([2]) that a matrix 
is 
if and only if it can be written in the form 
with 
unitary and 
nonsingular. A square complex matrix 
is called 
if it can be written in the form 
where 
is fixed unitary and 
is arbitrary matrix in
. To our knowledge, so far there has been little investigation of this 
solution to (2).
Motivated by the work mentioned above, we investigate 
solution to (2). We also consider the optimal approximation problem
(3)
where 
is a given matrix in 
and 
the set of all 
solutions to (2). In many case Equation (2) has not an 
solution. Hence we need to further study its least squares solution, which can be described as follows: Let 
denote the set of all 
matrices with fixed unitary matrix 
in 
Find 
such that
(4)
In Section 2, we present necessary and sufficient conditions for the existence of the 
solution to (2), and give an expression of this solution when the solvability conditions are met. In Section 3, we derive an optimal approximation solution to (3). In Section 4, we provide the least squares 
solution to (4).
2. 
Solution to (2)
In this section, we establish the solvability conditions and the general expression for the 
solution to (2).
Throughout we denotes 
the set of all 
matrices with fixed unitary matrix 
in 
i.e.,
where 
is fixed unitary and 
is arbitrary matrix in
.
Lemma 2.1. ([3]) Let 
Then the system of matrix equations 
is consistent if and only if
In that case, the general solution of this system is
where 
is arbitrary.
Now we consider the 
solution to (1). By the definition of 
matrix, the solution has the following factorization:
Let
where 
then (2) has 
solution if and only if the system of matrix equations
is consistent. By Lemma 2.1, we have the following theorem.
Theorem 2.2. Let 
and
where 
Then the matrix Equation (2) has a 
solution in 
if and only if
(5)
In that case, the general 
solution of (1) is
(6)
where 
is arbitrary.
3. The Solution of Optimal Approximation Problem (3)
When the set 
of all 
solution to (2) is nonempty, it is easy to verify 
is a closed set. Therefore the optimal approximation problem (3) has a unique solution by [13]. We first verify the following lemma.
Lemma 3.1. Let 
Then the procrustes problem
has a solution which can be expressed as
where 
are arbitrary matrices.
Proof. It follows from the properties of Moore-Penrose generalized inverse and the inner product that
Hence,
if and only if
It is clear that 
with 
are arbitrary is the solution of the above procrustes problem.
Theorem 3.2. Let 
and
(7)
where 
Assume 
is nonempty, then the optimal approximation problem (3) has a unique solution 
and
(8)
Proof. Since 
is nonempty, 
has the form of (6). It follows from (7) and the unitary invariance of Frobenius norm that
Therefore, there exists 
such that the matrix nearness problem (3) holds if and only if exist 
such that
According to Lemma 3.1, we have
where 
are arbitrary. Substituting 
into (6), we obtain that the solution of the matrix nearness problem (3) can be expressed as (8).
4. The Least Squares 
Solution to (4)
In this section, we give the explicit expression of the least squares 
solution to (4).
Lemma 4.1. ([12]) Given 
Then there exists a unique matrix 
such that
And 
can be expressed as
where 
Theorem 4.2. Let 
and
where
, 
Assume that the singular value decomposition of 
are as follows
(9)
where 
and 
are unitary matrices, 
, 
Then 
can be expressed as
(10)
where 
and 
is an arbitrary matrix.
Proof. It yields from (9) that
Assume that
(11)
Then we have
Hence
is solvable if and only if there exist 
such that
(12)
(13)
It follows from (12) and (13) that
(14)
(15)
where 
Substituting (14) and (15)
into (11), we can get the form of elements in 
is (10).
Theorem 4.3. Assume the notations and conditions are the same as Theorem 4.2. Then
if and only if
(16)
where 
Proof. In Theorem 4.2, it implies from (10) that
is equivalent to 
has the expression (10)
with 
Hence (16) holds.
5. 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).