ABSTRACT

For, if the sum of the elements in each row and the sum of the elements in each column are both equal to 0, then A is called an indeterminate admittance matrix. If A is an indeterminate admittance matrix and a Hermitian matrix, then A is called a Hermitian indeterminate admittance matrix. In this paper, we provide two methods to study the least squares Hermitian indeterminate admittance problem of complex matrix equation, and give the explicit expressions of least squares Hermitian indeterminate admittance solution with the least norm in each method. We mainly adopt the Moore-Penrose generalized inverse and Kronecker product in Method I and a matrix-vector product in Method II, respectively.

Keywords:

Matrix Equation, Least Squares Solution, Least Norm Solution, Hermitian Indeterminate Admittance Matrices

1. Introduction

Firstly, we state some symbols that are used in this paper. The set of all real column vectors with n coordinates by, and the set of all real matrices by are denoted. Let and stand for the set of all real symmetric matrices and the set of all real anti-symmetric matrices, respectively. The set of all complex matrices is denoted by, and stands for the set of all Hermitian matrices. For, if the sum of the elements in each row and the sum of the elements in each column are both equal to 0, then A is called an indeterminate admittance matrix. is denoted to be the set of all indeterminate admittance matrices. For, if A is not only an indeterminate admittance matrix, but also a symmetry matrix, then A is called a symmetry indeterminate admittance matrix. Similarly, for, A stands for an anti-symmetric indeterminate admittance matrix if A is an indeterminate admittance matrix and an anti-symmetric matrix. and are denoted to be the set of all symmetry indeterminate admittance matrices and the set of all anti-symmetric indeterminate admittance matrices, respectively. For, if A is also a Hermitian matrix, then A is called a Hermitian indeterminate admittance matrix. is denoted to be the set of all Hermitian indeterminate admittance matrices. The transpose matrix, the conjugate transpose matrix and the Moore-Penrose generalized inverse of matrix A are denoted by, and, respectively. The identity matrix of order n is denoted by. The trace of matrix is

where is the jth column of the identity matrix. The 2-norm of the vector x by is denoted. For, we define the inner product:, then is a Hilbert inner product space and the norm of a matrix generated by this inner product is the matrix Frobenius norm.

Definition 1 ( [1] ). For matrix, let, and denote the following vector by:

(1)

Definition 2 ( [1] ). For matrix, let, , , , , and denote the following vector by:

(2)

Definition 3 ( [1] ). For matrix, let, , , , and denote the following vector by:

(3)

It is well known that indeterminate admittance matrices play important roles in circuit modeling and lattices network and so on [2] [3] . In this paper, we mainly discuss the least squares problem associated with indeterminate admittance matrices, and derive it as follows.

Problem I. Given, , , , , and, let

Find such that

(4)

The solution is also called the least squares Hermitian indeterminate admittance solution of complex matrix equation

(5)

with the least norm.

For studying Problem I mentioned above, we first state some Lemmas.

Lemma 1. ( [4] ) The matrix equation, with and, has a solution if and only if

(6)

in this case it has the general solution

(7)

where is an arbitrary vector.

Lemma 2. ( [4] ) The least squares solutions of the matrix equation, with and, can be represented as

(8)

where is an arbitrary vector, and the least squares solution with the least norm is.

Direct and iterative methods on solving the matrix equations associated with the constrained matrix (such as Hermitian matrix, anti-Hermitian matrix, bisymmetric matrix, reflexive matrix) sets have been widely investigated. See [5] - [25] and references cited therein. Yuan, Liao and Lei [1] derived the least squares symmetric solution with the least norm of real matrix equation by using the vec-operator, Kronecker product and the Moore-Penrose generalized inverse. In order to avoid the difficulties of the coefficient matrices with large size from the Kronecker product, Yuan and Liao [26] recently improved this method, defined a matrix-vector product, and successfully carried out a special vectorization of the matrix equation to derive the least squares Hermitian solution with the least norm. Based on these methods, we continue to study Problem I in this paper.

We now briefly introduce the contents of our paper. In Section 2, by using the Moore-Penrose generalized inverse and the Kronecker product, we derive the least squares Hermitian indeterminate admittance solution with the least norm for the complex matrix Equation (5). In Section 3, we firstly discuss a class of linear least squares problem in Hilbert inner product, and analysis a matrix-vector product of. Then we present the explicit expression of the solution for the complex matrix Equation (5) by using the method.

2. Method I for the Solution of Problem I

In this section, we present the expression of the least square Hermitian indeterminate admittance solution of complex matrix Equation (5) with the least norm by using the Moore-Penrose generalized inverse and the Kronecker product of matrices.

Definition 4. For, , the symbol stands for the Kronecker product of A and B.

Theorem 3. Suppose and . Then

1) (9)

where is represented as (2), and the matrix is of the following form

(10)

2) (11)

where is represented as (3), and the matrix is of the following form

(12)

Proof. 1) For, X can be expressed as

It then follows that

Thus we have

Conversely, if the matrix satisfies, then it is easy to see that.

2) For, X can be expressed as

It then follows that

Thus we have

Conversely, if the matrix satisfies, then it is easy to see that. The proof is completed.

Theorem 4. Suppose, then

(13)

where and are represented as (2) and (3), and the matrix are in the forms (10) and (12).

Proof. For, then, we have

Thus we can get. Then and. By (9) and (11),

Conversely, if the matrix satisfies , then it is easy to see that. The proof is completed.

We now consider Problem I by using the Moore-Penrose generalized inverse and Kronecker product of matrices.

Theorem 5. Given, , , , , and, are defined as (10) and (12), are defined as (1), (2) and (3). Then the set of the problem can be expressed as

(14)

where

where y is an arbitrary vector.

Furthermore, the unique least squares Hermitian indeterminate admittance solution with the least norm can be expressed as

(15)

Proof. By Theorem 4, we can get

Thus, by Lemma 2,

By Theorem 2, it follows that

Thus we have

The proof is completed.

We now discuss the consistency of the complex matrix Equation (5). By Lemma 1 and Theorem 3, we can get the following conclusions.

Corollary 6. The matrix Equation (5) has a solution if and only if

(16)

In this case, denote by the solution set of (5). Then

(17)

Furthermore, if (16) holds, then the matrix Equation (5) has a unique solution if and only if

(18)

In this case,

(19)

The least norm problem

has a unique solution and can be expressed as (15).

3. Method II for the Solution of Problem I

The method for solving Problem I used in this section is from [26] . We concisely recall it as follows.

Definition 5. Let, and,. Define

1);

2).

Let, , and. By Definition 5, we have the following facts which are useful in this paper.

1);

2);

3);

4);

5);

6) is no meaning.

Suppose, , , , , ,. Then

7);

8);

9);

10).

Suppose, , , ,. Then

11).

Suppose, , ,.. Then

12);

13).

Lemma 7. ( [26] ) Given matrices and, let

(20)

Let that satisfies

(21)

If the matrix Equation (21) is consistent, then the solution set of the matrix Equation (21) is exactly the solution set of the following consistent system

(22)

Lemma 8. ( [26] ) Given and the matrices, let

such that

(23)

Then the solution set of (23) is the solution set of the system (22).

We now analyze the structure of the complex matrix equation over with the new product that we have presented.

Let

(24)

where

Let

(25)

Note that.

Let

(26)

Note that. We can get the following lemmas.

Lemma 9. Suppose, then

(27)

where is represented as (2), and the matrix is in the form (25).

Lemma 10. Suppose, then

(28)

where is represented as (3), and the matrix is in the form (26).

Lemma 11. Suppose, then

(29)

where and are represented as (2) and (3). The matrix are in the form (25) and (26).

Theorem 12. Suppose, , , and. Let and, where is the ith column vector of matrix A, and is the ith column vector of matrix C,

(30)

where is the jth row vector of matrix B, and is the jth row vector of matrix D. Then

1) (31)

2) Let, where

Thus

(32)

(33)

3) Let, where

Thus

(34)

(35)

Proof. 1)

2) By (1), Definition 5 and Lemma 7, we can get

3) The proof is similar to that of (2), so we omit it.

The proof is completed.

We now use Lemmas 7 - 11, and Theorem 12 to consider the least squares Hermitian indeterminate admittance solution for the matrix Equation (5). The following notations and lemmas are necessary for deriving the solutions.

For, , , , , and, let

(36)

where,

Theorem 13. Let, , , , , and, are defined as (25) and (26), are defined as (2) and (3). let be as in (36). Then can be expressed as

(37)

where y is an arbitrary vector.

Furthermore, the unique least squares Hermitian indeterminate admittance solution with the least norm can be expressed as

(38)

Proof. By Theorem 4, we can get

Then by Lemma 11, the least squares problem

with respect to the Hermitian indeterminate admittance matrix X is equivalent to the following consistent matrix equation

Thus, by Lemma 2, if and only if

From Lemma 11, it follows that

where y is an arbitrary vector. it yields that

The proof is completed.

We now discuss the consistency of the complex matrix Equation (5). By Lemma 1 and Theorem 13, we can get the following conclusions.

Corollary 14. The matrix Equation (5) has a solution if and only if

(39)

In this case, denote by the solution set of (5). Then

(40)

where y is an arbitrary vector.

Furthermore, if (39) holds, then the matrix Equation (5) has a unique solution if and only if

(41)

In this case,

(42)

The least norm problem

has a unique solution and can be expressed as (38).

4. Conclusion

In this paper, we mainly consider the least squares Hermitian indeterminate admittance problem of the complex matrix equation. We derive the explicit solution of this complex matrix equation over The paper provide a direct method to solve the least squares admittance problem of complex matrix equation. More works such as iterative methods, error analysis and numerical stability need to be investigated in future.

Funding

The research is supported by Natural Science Foundation of China (No. 11571220), Guangdong Natural Science Fund of China (No. 2015A030313646), and the Characteristic Innovation Project (Natural Science) of the Education Department of Guangdong Province (No. 2015KTSCX148).

Cite this paper

Liang, Y.F., Yuan, S.F., Tian, Y. and Li, M.Z. (2018) Least Squares Hermitian Problem of Matrix Equation (AXB, CXD) = (E, F) Associated with Indeterminate Admittance Matrices. Journal of Applied Mathematics and Physics, 6, 1199-1214. https://doi.org/10.4236/jamp.2018.66101

References