A Solution Matrix by IEVP under the Central Principle Submatrix Constraints

Bhatt, Vineet; Kaur, Manpreet; AL-Hammadi, I. Ahmed M.; Kumar, Sunil

doi:https://doi.org/10.1155/2024/7908231

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 7908231 | https://doi.org/10.1155/2024/7908231

A Solution Matrix by IEVP under the Central Principle Submatrix Constraints

Vineet Bhatt,¹Manpreet Kaur,²I. Ahmed M. AL-Hammadi,³and Sunil Kumar¹

Academic Editor: Ram Jiwari

Received18 Oct 2022

Revised13 Aug 2023

Accepted06 Apr 2024

Published06 May 2024

Abstract

The real matrix is called centrosymmetric matrix if where is permutation matrix with ones on cross diagonal (bottom left to top right) and zeroes elsewhere. In this article, the solvability conditions for left and right inverse eigenvalue problem (which is special case of inverse eigenvalue problem) under the submatrix constraint for generalized centrosymmetric matrices are derived, and the general solution is also given. In addition, we provide a feasible algorithm for computing the general solution, which is proved by a numerical example.

1. Introduction

Linear algebra plays an important role in different fields of engineering and sciences. Especially matrix algebra has been used in many fields of applications since the 19^th century [1, 2]. A symmetric matrix is a well-known type of matrix that also possesses numerous applications [3, 4]. One of the famous types of a symmetric matrix is the Centro-symmetric matrix (also known as cross-symmetric matrix) [5]. These matrices are symmetric about their center and play an important role in several applications such as antenna theory, vibration in structure, detection, estimation, electrically network engineering, communication sciences, predication theory, speech analysis, quantum physics, etc. [3, 4, 6–13]. The study of eigenvalues and eigenvectors plays an important role in enhancement in the fields of applications of Centro-symmetric matrices. Eigenvalues are a particular set of scalars associated with the system of linear equations whereas, eigenvectors are vectors that are changed by scalars when the linear transformation is applied [14, 15]. Symmetric matrices are an unusual structure that helps with main component extraction and determinant assessment by reducing the computational complexity [12]. For ordered Centro-symmetric matrices, the issue of principal component extraction may be simplified into two sub-problems of principal component assessment with orders for even and for odd. The multiplicative complexity associated with the computation of the determinant and main components is reduced roughly by 75% [12]. An algorithm for the evaluation of eigenvalues and eigenvectors for centro-symmetric matrices with numerical solutions has been discussed [13]. Currently, many researchers have been working on special types of eigenvalues problems known as IEP (Inverse eigenvalues problem) [16]. IEP concerns the reconstruction of a structured matrix from prescribed spectral data, which has been used in control theory, vibration theory, structural design, molecular spectroscopy, etc. In the case of Centro-symmetric potentials, authors [17] explicitly solved IEP for classical Liouville functions in the term of Hermitian characteristics functions and found that eigenvalues were related to the constant of motion. Furthermore [17], obtained the same results for the Wigner distribution function by using the semi-classical quantum wave function. The results related to the eigenvector and eigenvalues for per-symmetric matrices occurred in communication and information theories have been already discussed [18]. In [19], the authors studied quadratic IEP for dumped structural updating model and developed a new approach without using eigenvector expansion techniques. In [20, 21], the natural frequencies of vibration of a beam of a given length in the free configuration were found by solving the Eigenvalue problem. On the bases of the solution of IEP, the method was developed for the design of the structure with low-order natural frequencies [22]. IEP is also applicable for finding the solution to problems related to molecular spectroscopy [23]. In [24], the authors modified various quadratically convergent methods for solving IEVP with consideration of both cases such as distinct and multiple eigenvalues. While studying the papers, it has been observed that many authors studied IEVP for Centro-symmetric matrices under the submatrix constraint [25–29]. But few authors studied left and right IEVP for Centro-symmetric matrix under submatrix constraint [30–32]. Therefore, we will consider the problem of the left and right IEVP of the Centro-symmetric matrix with central principal submatrix constraint. In this paper, we will consider that the extend matrix and submatrix both have the same structure ad also both are centro-symmetric matrices. This paper is divided into 4 sections. The first part is introductory, notations and some important definitions are in the second part. In the third part, we provide existence and expression for solution of the inverse eigenvalue problem for centrosymmetric matrices under central principal submatrix constraint and conclusion included in fourth part.

2. Notations and Preliminaries

In this article, we consider and be the sets of all orthogonal matrices and ) the real matrices respectively. , and denotes the Moore-Penrose generalized inverse, rank, and transpose of matrix respectively. An order identity matrix, reverse identity matrix and zero matrices are represented by , and respectively. For any two matrices = and = , the Hadamard and inner product are represented by = and = trace (), respectively. represented as Hilbert space and notation used for Frobenius norm of matrix is = . The scalars and denotes the left and right eigenvalues of matrix and and be left and right eigenvectors corresponding these eigenvalues.

2.1. Basic Definitions for Centro-Symmetric Matrix

In this section, we present some important definitions for Centro-symmetric matrices with some appropriate examples. Definition 1 is defined for the construction of the Centro-symmetric matrix and Definition 2 for finding the central principal submatrix and trailing principal submatrix, Definition 3 for orthogonality, and Definition 4 for left and right eigenpairs. Similarly, Definition 5 for symmetric and antisymmetric vectors.

Definition 1. The centro-symmetric matrix is defined bywhere are natural numbers. denote the set of all Centro-symmetric matrices. For example, is a Centro-symmetric matrix of order 3.
Several researchers have shown great interest in the study of inverse eigenvalue problems under submatrices constraint [25, 27, 28, 33]. Due to the unique structure of the Centro-symmetric matrix, it is not suitable for discussing Centro-symmetric matrices under principal submatrices constraints, for it destroys the symmetry in the structure of Centro-symmetric matrices. Therefore, we discuss the different concepts like the central principal submatrix, which is first defined in [25].

Definition 2. An m-square central principal matrix of the matrix is given bywhere is a zero matrix of order , and is an identity matrix of order m.
For example, for the matrix P is of order 4, then isHere, is the central principal submatrix of order 2.
And for the matrix P is of order 5, then isHere, is the central principal submatrix of order 3.
From above these two examples, it is clearly shown that the submatrix is lie in the center of the matrix, and matrices of order even (odd) have only submatrices of order even (odd). The central principal submatrix is also a centrosymmetric matrix.

2.1.1. Trailing Principal Submatrix

A m-square trailing principal submatrix of real matrix ifwhere is a zero matrix of order , and is an identity matrix of order m.

For example, firstly, consider the matrix is of even order given in (3), then using (5), we get is trailing principal submatrix of order 2.

Secondly, consider the matrix is of odd order given in (4), then using (5), we get is trailing principal submatrix of order 3.

From above these two examples, it is clearly shown that the submatrix is situated in the left corner of the matrix, and matrices of order even (odd) have only submatrices of order even (odd). The trailing principal submatrix is not centrosymmetric matrix.

Definition 3. If matrix of order n satisfies , (where is an identity matrix of order ), then the matrix is called an orthogonal matrix [34].

Definition 4. A pair involved eigenvalue and its corresponding eigenvector i.e., , where is an eigenvalue and is corresponding eigenvector is known as an eigen-pair. For partial right eigenpairs and left eigenpairs , to construct a matrix , such thatwhere , are eigenvalues, and , are their corresponding eigenvectors, and S is a subspace of . In this paper, we discuss the problem and its optimal approximation for the centro-symmetric matrix under central principal submatrix constraint that is in the light of extended matrix which preserves the Centro-symmetric property. Now, we discuss the special structure of eigenvalues and their corresponding eigenvectors for a real matrix. If matrix has real right eigenpairs , , where are real numbers), then, let . If matrix has real right eigenpairs , , where are real numbers, ), then, let . Hence, the equation (6) becomesSimilarly, if are left real eigenpairs of , then, let . If are left complex eigenpairs of , then, let , and Now, (5) becomeFor right eigenpairs , , and left eigenpairs , writeHence, equations (8) and (9) become

Definition 5. Let is a symmetric vector if is an anti-symmetric vector if .

Problem 6. If , , , and expressed in (8), and be submatrix with Centro-symmetric structure, find an extended matrix such that

3. Preliminary Lemmas and General Solutions to Problem 6

In this section, we discuss the central submatrix having the same symmetric properties and structure as the Centro-symmetric matrix. Therefore, they have similar expressions. Furthermore, we discuss the properties of eigenvalues and eigenvectors of the centro-symmetric matrix, and we expressed the special form of eigenvectors of the centro-symmetric matrix. Hence, using a special form of centro-symmetric matrix and its central principal submatrix, we convert Problem 6 into two inverse eigenvalue problems of half-sized independent real matrices under principal submatrices constraint. Furthermore, we provide necessary and sufficient conditions for the existence of a solution to Problem 6 and give an expression for the general solution.

Now, be column of , and let , , …, , , then Let where is the greatest integer less than or equal to , and let orthogonal matrices:

Lemma 7 (see [35]). A matrix is a Centro-symmetric matrix of order iff

Lemma 8 (see [25]). A matrix if and only if there exist such that

Proof. When , consider , If , then by Lemma 7
,which is equivalent to
Hence,Let , then
Conversely, for every , we have.From above, it is showing that it follows Lemma 7. Hence, .
Now, we discuss the special properties of the central principal submatrix of order of the centro-symmetric matrix in which the submatrix has the same structure and symmetric properties as those given in the centro-symmetric matrix. Here, we assume , and , where r and t are the greatest integer less than or equal to and respectively.

Lemma 9. Assume that have formed as in equation (12). Then the k-square central principal submatrix of is given as

Proof. If , from (12) and properties that are, a matrix of even order only having central principal submatrices of even order, so we have , and
, where
Thus, and set . Hence, the -square central principal submatrix of may be expressed asIf , and a matrix of odd order has central principal submatrices of odd order, we have , andwhere ,
Hence,By setting, , then the may be written asBy combining (18) and (20), we get -square central principal submatrix of that has the form as in (16).

Lemma 10. Let have formed as in (12). Then the k-square central principal submatrix of P is given aswhere . The matrix is central principal both are trailing principal submatrix of and , respectively.

Given , if (where ) are right and left real eigenpairs respectively, then we get, from Lemma 7,

Therefore, are eigenvectors associated with , where is a symmetric vector, while is an anti-symmetric vector. Similarly, is a symmetric vector, and is an anti-symmetric vector.

If (where ) are right and left complex eigenpairs respectively, then we get, from Lemma 7,

Thus where the columns of are symmetric vectors, and . Similarly, the columns of are symmetric vectors, and .

From the above analysis, without loss of generality, we may suppose that , and in Problem 6 can be written as follows:where, , , where are block diagonals. Thus, and has the following form:

If , then

If then

Now, for , set , and for set , then for all arbitrary and may be written in the following form:where .

Lemma 11. Given and as in (8), then there exists a matrix such that

If and only if

Moreover, its general solutions can be written as follows:where .

Lemma 12 (see [25]). Assume that be given. Denote

Then, every element of has following form:

In particular, has matrices solutions in , iff , and its general solution can be also expressed in the form of (32).

Theorem 13. Given partition as in (16). Let be given as in (17), and be given as in (18). Partition and as in (19). Denotewhere,

Then, Problem 6 is solvable if and only if

Furthermore, every matrix in the solution set may be expressed as follows:where, and and be any arbitrary (38).

Proof. From Lemmas 8, 9, and Problem 6 is equivalent to evaluating and , such thatwhere and satisfyBy Lemma 11, we know that the equations in (38) hold if and only ifwhich means that (35) hold. Furthermore, and can be expressed aswhere and be an arbitrary real matrix.
Now, the definitions of as in (33), we substitute (42) into (40), thenLemma 12 implies that (43) holds if and only ifwhich means (36) holds, and may be written aswhere, and be arbitrary.
Therefore, the solution to Problem 6 has the form of (37)The above matrix is the solution of Problem 6, which satisfies equation (11) and centrosymmetric in nature. The conditions for solvability of Problem 6 given in equation (35), which are very useful conditions for finding the proof of Theorem 13 i.e., solution of problem-1. In above proof it is clearly shown that the left and right eigenpairs of matrix and central principal submatrix of solution matrix satisfies conditions of the Problem 6. In Problem 6, if , then this problem becomes Problem 6 in [25]. (see Algorithm 1).

(i)	Input as in (24), as in (25), and .
(ii)	Partition as in (21) and get , , and get as in (25), and derive as in (28).
(iii)	Construct as in (22) and (23).
(iv)	Obtain by using (33).
(v)	If (35) and (36) hold, then go to the next step, otherwise reconstruct from step III.
(vi)	Form (38), calculate .
(vii)	Follows (37), obtain matrix .

Example 1. Let n = 8 m = 4, l = 3, k = 4, and given , and , where , , and Equation (33) becomes.From equation (37), the general solution is.where is a general solution to Problem 6, which is a centro-symmetric matrix and is a central principle submatrix.

Example 2. Let n = 10, m = 5, l = 4, k = 4, and given , and , whereEquation (33) becomesFrom equation (37), the general solution is.where is a general solution to Problem 6, which is a centro-symmetric matrix and is a central principle submatrix.

Example 3. Let n = 10, m = 5, l = 4, k = 4, and given , and , where Equation (33) becomesFrom equation (37), the general solution isFrom above numerical example, the resultant matrix is centrosymmetric matrix, which having as central principle submatrix.

4. Conclusion

In this paper, we expanded the system from the center subsystem satisfying the matrix constraint, where and are both centrosymmetric matrices. We consider the left and right inverse eigenvalue problem with a central principal submatrix constraint, which lies in center of the original matrix. Using special properties of left and right eigenpairs, we obtained the solvability conditions of the Problem 6 and its general solution. Furthermore, for feasibility of obtained general centrosymmetric matrix solutions we provided an algorithm with numerical example. In future, we will discuss the unique solution to the optimal approximation problem.

Data Availability

The data presented in this research article are available on request from the corresponding author.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors have contributed to all parts of the article. All authors have read and approved the final manuscript.

Acknowledgments

The authors express their hearty thanks and gratefulness to all those scientists whose masterpieces have been consulted during the preparation of the present research article.

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Copyright © 2024 Vineet Bhatt et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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