ABSTRACT

A square complex matrix is called if it can be written in the form with being fixed unitary and being arbitrary matrix in. We give necessary and sufficient conditions for the existence of the solution to the system of complex matrix equation and present an expression of the solution to the system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least square solution with least norm to this system mentioned above is considered. The representation of such solution is also derived.

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).

REFERENCES