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For <i> A</i>∈<B>C</B><sup>mΧn</sup> , 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 (<i>AXB,CXD</i>)=(<i>E,F</i>) , 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.

Firstly, we state some symbols that are used in this paper. The set of all real column vectors with n coordinates by

where

Definition 1 ( [

Definition 2 ( [

Definition 3 ( [

It is well known that indeterminate admittance matrices play important roles in circuit modeling and lattices network and so on [

Problem I. Given

Find

The solution

with the least norm.

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

Lemma 1. ( [

in this case it has the general solution

where

Lemma 2. ( [

where

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 [

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

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

Theorem 3. Suppose

1)

where

2)

where

Proof. 1) For

It then follows that

Thus we have

Conversely, if the matrix

2) For

It then follows that

Thus we have

Conversely, if the matrix

Theorem 4. Suppose

where

Proof. For

Thus we can get

Conversely, if the matrix

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

Theorem 5. Given

where

where y is an arbitrary vector.

Furthermore, the unique least squares Hermitian indeterminate admittance solution with the least norm

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

In this case, denote by

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

In this case,

The least norm problem

has a unique solution

The method for solving Problem I used in this section is from [

Definition 5. Let

1)

2)

Let

1)

2)

3)

4)

5)

6)

Suppose

7)

8)

9)

10)

Suppose

11)

Suppose

12)

13)

Lemma 7. ( [

Let

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

Lemma 8. ( [

such that

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

We now analyze the structure of the complex matrix equation

Let

where

Let

Note that

Let

Note that

Lemma 9. Suppose

where

Lemma 10. Suppose

where

Lemma 11. Suppose

where

Theorem 12. Suppose

where

1)

2) Let

Thus

3) Let

Thus

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

where

Theorem 13. Let

where y is an arbitrary vector.

Furthermore, the unique least squares Hermitian indeterminate admittance solution with the least norm

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,

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

In this case, denote by

where y is an arbitrary vector.

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

In this case,

The least norm problem

has a unique solution

In this paper, we mainly consider the least squares Hermitian indeterminate admittance problem of the complex matrix equation

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).

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