The Solutions to Matrix Equation with Some Constraints
Let be a given Hermitian matrix satisfying . Using the eigenvalue decomposition of , we consider the least squares solutions to the matrix equation with the constraints and . A similar problem of this matrix equation with generalized constrained is also discussed.
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.
It is a very active research topic to study solutions to various matrix equations [1–4]. There are many authors who have investigated the classical matrix equation 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 , and Don  have presented the existence conditions and detailed representations of constrained solutions for (1) with corresponding constraints, respectively. For instance, Peng and Hu  presented the eigenvectors-involved solutions to (1) with reflexive and antireflexive constraints; Wang and Yu  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  proposed an eigenvectors-free method to (1) with and constraints, where is a Hermitian involutory matrix and .
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
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.
2. 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 where . Partitioning (4) is equivalent to Therefore, The constraint is equivalent to with .
Partition and denote then assume that the singular value decomposition of and is as follows: where , and are unitary matrices, , , , , , and .
Theorem 1. Given . Then the least squares solutions to the matrix equation (1) with the constraints and can be expressed as where and 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 Hence is solvable if and only if there exist such that It follows from (19) that Substituting (20) into (16) and then into (8), we can get that the form of is (11).
3. Least Squares Solutions to the Matrix Equation (1) with the Constraints and
In this section, we generalize the constraints to , where is a given unitary matrix of order . Obviously, the constraint is equal to Notice that (1) can be equivalently rewritten in Denoting by and setting , the equation becomes with the constraints and .
Theorem 2. Given . Then the least squares solutions to the matrix equation (1) with the constraints and can be expressed aswhere and are arbitrary matrix.
4. An Algorithm and Numerical Examples
Based on the main results of this paper, we in this section propose an algorithm for finding the least squares solutions to the matrix equation with the constraints and . All the tests are performed by MATLAB 6.5 which has a machine precision of around .
Example 4. Suppose
Applying Algorithm 3, we obtain the following:
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
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).