A Solution Matrix by IEVP under the Central Principle Submatrix Constraints

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Introduction
Linear algebra plays an important role in diferent felds of engineering and sciences.Especially matrix algebra has been used in many felds 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].Tese 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][7][8][9][10][11][12][13].Te study of eigenvalues and eigenvectors plays an important role in enhancement in the felds 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 n ordered Centro-symmetric matrices, the issue of principal component extraction may be simplifed into two subproblems of principal component assessment with orders n/2 for even and (n − 1)/2 and (n + 1)/2 for odd.Te 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.Te 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 confguration 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 fnding the solution to problems related to molecular spectroscopy [23].In [24], the authors modifed 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][26][27][28][29].But few authors studied left and right IEVP for Centro-symmetric matrix under submatrix constraint [30][31][32].Terefore, 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.Tis paper is divided into 4 sections.Te frst part is introductory, notations and some important defnitions 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.

Notations and Preliminaries
In this article, we consider O and R m×n be the sets of all (n × n) orthogonal matrices and (m × n) the real matrices respectively.P + , ρ(P) and P T denotes the Moore-Penrose generalized inverse, rank, and transpose of matrix P respectively.An n th order identity matrix, reverse identity matrix and zero matrices are represented by I n , S n and 0 respectively.For any two matrices P � (p i,j ) and Q � (q i,j ) ∈ R m×n , the Hadamard and inner product are represented by P * Q � (p i,j q i,j ) and 〈P, Q〉 � trace (Q T P), respectively.R m×n represented as Hilbert space and notation used for Frobenius norm of matrix P is ‖P‖ � (trace(P T P)) 1/2 .Te scalars λ and μ denotes the left and right eigenvalues of matrix and x and y be left and right eigenvectors corresponding these eigenvalues.

Basic Defnitions for Centro-Symmetric Matrix.
In this section, we present some important defnitions for Centrosymmetric matrices with some appropriate examples.Defnition 1 is defned for the construction of the Centrosymmetric matrix and Defnition 2 for fnding the central principal submatrix and trailing principal submatrix, Definition 3 for orthogonality, and Defnition 4 for left and right eigenpairs.Similarly, Defnition 5 for symmetric and antisymmetric vectors.Defnition 1. Te centro-symmetric matrix P � (p i,j ) ∈ R n×n is defned by where i, j are natural numbers.CSR n×n denote the set of all (n × n) Centro-symmetric matrices.For example, 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.Terefore, we discuss the different concepts like the central principal submatrix, which is frst defned in [25].
Defnition 2. An m-square central principal matrix P c (m) of the matrix P, is given by where 0 is a zero matrix of order (m × ((n − m)/2)), and I is an identity matrix of order m.For example, for the matrix P is of order 4, then P c (2) is is the central principal submatrix of order 2.
And for the matrix P is of order 5, then P c (3) is Here, 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).Te central principal submatrix is also a centrosymmetric matrix.

Trailing Principal Submatrix.
A m-square trailing principal submatrix P t (m) of real matrix P if where 0 is a zero matrix of order (m × (n − m)), and I is an identity matrix of order m.

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For example, frstly, consider the matrix P is of even order given in (3), then using (5), we get is trailing principal submatrix of order 2. Secondly, consider the matrix P is of odd order given in (4), then using (5), we get P t (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).Te trailing principal submatrix is not centrosymmetric matrix.
(where I is an identity matrix of order n), then the matrix is called an orthogonal matrix [34].Defnition 4. A pair involved eigenvalue and its corresponding eigenvector i.e., (λ, x), where λ is an eigenvalue and x is corresponding eigenvector is known as an eigenpair.For partial right eigenpairs (λ i , x i ), i � 1, 2, . . ., h 1 , and left eigenpairs (μ j , y j ), j � 1, 2, . . ., h 2 , to construct a (n × n) matrix P ∈ S, such that where h 1 ≤ n, h 2 ≤ n, λ i , μ j are eigenvalues, and x i , y j are their corresponding eigenvectors, and S is a subspace of R n×n .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 P matrix has real right eigenpairs (λ i , x i ), i � 1, 2, . . ., h 1 , where Hence, the equation ( 6) becomes Similarly, if (μ j , y j ), j � 1, 2, . . ., h 2 are left real eigenpairs of P, then, let  Γ j � μ j ,  Y j � y j .If (μ j , y j ), j � 1, 2, . . ., h 2 are left complex eigenpairs of P, then, let For right eigenpairs (λ i , x i ), i � 1, 2, . . ., h 1 , and left eigenpairs (μ j , y j ), j � 1, 2, . . ., h 2 , write Hence, equations ( 8) and ( 9) become x is an anti-symmetric vector if S n x � −x.

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 Centrosymmetric matrix.Terefore, 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 centrosymmetric matrix.Hence, using a special form of centrosymmetric 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 sufcient conditions for the existence of a solution to Problem 6 and give an expression for the general solution.Now, e i be i th (i ∈ natural numbers) column of I n , and let S n � (e n , e n−1 , . .., e 2 , e 1 ), then , where [n/2] is the greatest integer less than or equal to n/2, and let orthogonal matrices: Lemma 7 (see [35]).A matrix P is a Centro-symmetric matrix of order n if S n PS n � P.
Lemma 8 (see [25]).A matrix P ∈ CSR n×n if and only if there exist Hence, Let P 1 � P 11 + P 21 S k , P 2 � P 11 − P 21 S k , then P � Conversely, for every P 1 , P 2 ∈ R k×k , we have.
From above, it is showing that it follows Lemma 7.
Hence, � D n P 1 0 0 P 2  D T n ∈ CSR n×n .Now, we discuss the special properties of the central principal submatrix of order k 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 � [k/2] , and r � [n/2], where r and t are the greatest integer less than or equal to k/2 and n/2 respectively.□ Lemma 9. Assume that P ∈ CSR n×n have formed as in equation (12).Ten the k-square central principal submatrix of P is given as Proof.If n � 2r, from (12) and properties that are, a matrix of even order only having central principal submatrices of even order, so we have k � 2t, and Hence, the k-square central principal submatrix of P may be expressed as If n � 2r + 1, and a matrix of odd order has central principal submatrices of odd order, we have k � 2t + 1, and where M, N ∈ R t×t , u t � (0, I t )u, v T t � (0, I t )v.Hence, By setting, , then the P c (k) may be written as 4

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By combining (18) and ( 20), we get k-square central principal submatrix of P that has the form as in (16).□ Lemma 10.Let P ∈ CSR n×n have formed as in (12).Ten the k-square central principal submatrix of P is given as where P 10 ∈ R (k−t)×(k−t) andP 20 ∈ R t×t .Te matrix P 0 (k) is central principal both are trailing principal submatrix of P 1 and P 2 , respectively.
Given P ∈ CSR n×n , if (λ i , x i ), (μ j , y j ) (where 1 ≤ i ≤ h 1 , 1 ≤ j ≤ h 2 ) are right and left real eigenpairs respectively, then we get, from Lemma 7, and y j T S n P � y j T PS n � μ j S n y j T , ( Terefore, x i ± S n x i are eigenvectors associated with λ i , where x i + S n x i is a symmetric vector, while x i − S n x i is an anti-symmetric vector.Similarly, y j T + S n y j T is a symmetric vector, and y j T − S n y j T is an anti-symmetric vector.If (λ i , x i ), (μ j , y j ) (where 1 ≤ i ≤ h 1 , 1 ≤ j ≤ h 2 ) are right and left complex eigenpairs respectively, then we get, from Lemma 7, and , where the columns of From the above analysis, without loss of generality, we may suppose that X, Y, and Γ, Λ in Problem 6 can be written as follows: where, , where Λ 1 , Λ 2 , Γ 1 , Γ 2 are block diagonals.Tus, D T n X and D T n Y has the following form: If n � 2r, then If n � 2r + 1, then

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Now, for n � 2r, set , then for all arbitrary n, D T n X and D T n Y may be written in the following form: where and Γ ∈ R l×l as in (8), then there exists a matrix P ∈ R n×n such that If and only if Moreover, its general solutions can be written as follows: where Lemma 12 (see [25]).Assume that X ∈ R m×m , Y ∈ R n×l , B ∈ R k×l be given.Denote Ten, every element of U 1 has following form: In particular, f 1 (P) � 0 has matrices solutions in R m×n , if X + XBYY + � B, and its general solution can be also expressed in the form of (32).
Theorem 13.Given P 0 ∈ CSR k×k , partition P 0 as in (16).Let X ∈ R m×m , and Y ∈ R n×l be given as in (17), Λ ∈ R m×m , and Γ ∈ R l×l be given as in (18).Partition D T n X and D T n Y as in (19).Denote where, Ten, Problem 6 is solvable if and only if (36) Furthermore, every matrix P in the solution set may be expressed as follows: where, , and G 3 ∈ R (n−r−r 3 )×(n−r−r 4 ) and G 4 ∈ R (r−r 5 )×(r−r 6 ) be any arbitrary (38).
Proof.From Lemmas 8, 9, and Problem 6 is equivalent to evaluating P 1 [n − r] and P 2 [r], such that where P 1 and P 2 satisfy (39) By Lemma 11, we know that the equations in (38) hold if and only if 6 Journal of Mathematics (41) which means that (35) hold.Furthermore, P 1 and P 2 can be expressed as where and G 2 ∈ R (r−r 5 )×(r−r 6 ) be an arbitrary real matrix.Now, the defnitions of K 1 , K 2 , H 1 , H 2 , H 3 , H 4 as in ( 33), we substitute (42) into (40), then Lemma 12 implies that (43) holds if and only if which means (36) holds, and G 1 , G 2 may be written as where, G 3 ∈ R (n−r−r 3 )×(n−r−r 4 ) and G 4 ∈ R (r−r 5 )×(r−r 6 ) be arbitrary.Terefore, the solution to Problem 6 has the form of (37) Te above matrix P is the solution of Problem 6, which satisfes equation (11) and centrosymmetric in nature.Te conditions for solvability of Problem 6 given in equation (35), which are very useful conditions for fnding the proof of Teorem 13 i.e., solution of problem-1.In above proof it is clearly shown that the left and right eigenpairs of matrix P and central principal submatrix of solution matrix P satisfes conditions of the Problem 6.In Problem 6, if Y � 0, then this problem becomes Problem 6 in [25].(see Algorithm 1).
where P is a general solution to Problem 6, which is a centrosymmetric matrix and P 0 is a central principle submatrix.
ALGORITHM 1: Left-right inverse eigenvalue problem for generalized centro-symmetric matrices.
8 Journal of Mathematics Journal of Mathematics Equation ( 33) becomes where P is a general solution to Problem 6, which is a centrosymmetric matrix and P 0 is a central principle submatrix.
From above numerical example, the resultant matrix P is centrosymmetric matrix, which having P 0 as central principle submatrix.

Conclusion
In this paper, we expanded the system P from the center subsystem P 0 satisfying the matrix constraint, where P and P 0 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.

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Proof.When n � 2k, consider P � P 11 P 12 P 21 P 22  , If P ∈ CSR n×n , then by Lemma 7 S k PS k � P 22 � S k P 11 S k � P 11 , P 12 � S k P 21 S k � P 21 . 2. where