On the bisymmetric nonnegative inverse eigenvalue problem

Somphotphisut, Somchai; Wiboonton, Keng

doi:10.1016/j.laa.2017.12.017

Cited by 3 publications

(5 citation statements)

References 15 publications

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“…Case 2. The list Γ ⌈ p 0 2 ⌉ is realizable by a centrosymmetric nonnegative matrix of order even: In this case we consider A as in (6), X as in (8) with all its columns normalized, and in ii) we take B = Ω + C. Then C = B − Ω, is centrosymmetric nonnegative, and by Lemma 1.1 and Theorem 1.3, the perturbation A+XCX T is centrosymmetric nonnegative with spectrum Λ.…”

Section: Remark 31 According To the Results In This Section It Is Cmentioning

confidence: 99%

“…In [8, Theorem 3.1] (see also [7]) it has been proved that the spectrum of a 3 × 3 nonnegative centrosymmetric matrix is real. Then, since bisymmetric matrices are centrosymmetric, we may use the necessary and sufficient conditions given in [3], for the existence of a 3 × 3 centrosymmetric nonnegative matrix with real eigenvalues λ 1 , λ 2 , λ 3 and diagonal entries ω 1 , ω 2 , ω 1 .…”

Section: Remark 31 According To the Results In This Section It Is Cmentioning

confidence: 99%

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Centrosymmetric nonnegative realization of spectra

Julio

Rojo

Soto

2019

Linear Algebra and its Applications

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A list Λ = {λ 1 , λ 2 , . . . , λ n } of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. In this paper we intent to characterize those lists of complex numbers, which are realizable by a centrosymmetric nonnegative matrix. In particular, we show that lists of nonnegative real numbers, and lists of complex numbers of Suleimanova type (except in one particular case), are always the spectrum of some centrosymmetric nonnegative matrix. For the general lists we give sufficient conditions via a perturbation result. We also show that for n = 4, every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix.{λ 1 , λ 2 , . . . , λ n }. If there exists an n × n nonnegative matrix A with spectrum Λ we say that Λ is realizable and that A is the realizing matrix. A complete solution for the NIEP is known only for n ≤ 4, which shows the difficulty of the problem. Throughout this paper, if Λ = {λ 1 , λ 2 , . . . , λ n } is realizable by a nonnegative matrix A, then λ 1 = ρ(A) = max{|λ i | , λ i ∈ Λ} is the Perron eigenvalue of A. We shall denote the transpose of a matrix A by A T , ⌊x⌋ and ⌈x⌉ denote the largest integer less than or equal to x and the smallest integer greater than or equal to x, respectively. J will denote the counteridentity matrix, that is, J = [e n | e n−1 | · · · | e 1 ]. Then it is clear that J T = J, J 2 = I. Observe that multiplying a matrix A by J from the left results in reversing its rows, while multiplying A by J from the right results in reversing its columns. A vector x is called symmetric if Jx = x.The set of all n × n real matrices with constant row sums equal to α ∈ R will be denote by CS α . It is clear that e = [1, 1, . . . , 1] T is an eigenvector of any matrix in CS α , corresponding to the eigenvalue α. The relevance of matrices with constant row sums is due to the well known fact that the problem of finding a nonnegative matrix A with spectrum Λ = {λ 1 , . . . , λ n } is equivalent to the problem of finding a nonnegative matrix B ∈ CS λ 1 with spectrum Λ, that is, A and B are similar if the Perron eigenvalue is simple and they are cospectral otherwise (see [4]).In this paper we study the NIEP for centrosymmetric matrices. Centrosymmetric matrices appear in many areas: physics, communication theory, differential equations, numerical analysis, engineering, statistics, etc. Now we state the definition and certain properties about centrosymmetric matrices.Definition 1.1 A matrix C = (c ij ) ∈ M m,n is said to be centrosymmetric, if its entries satisfy the relationThe definition means that a centrosymmetric matrix C can be written as J m CJ n = C, where J n is the n × n counteridentity matrix. That is, J n = [e n | e n−1 | · · · | e 1 ] . For m = n, we shall write J instead J n .

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Section: Remark 31 According To the Results In This Section It Is Cmentioning

confidence: 99%

Section: Remark 31 According To the Results In This Section It Is Cmentioning

confidence: 99%

Centrosymmetric nonnegative realization of spectra

Julio

Rojo

Soto

2019

Linear Algebra and its Applications

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show abstract

“…where (13) and (14), respectively; then, there exist orthogonal matrices Q * 1 and Q * 2 and inverse matrices Y 1 and Y 2 , such that…”

Section: The Solution Of Optimal Approximation Problemmentioning

confidence: 99%

“…Step 3. Calculate the singular value decomposition X D 1 , X D 2 as equations (13) and (14), verify that equation (24) is true and A can be expressed as equation (25 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , r � 0.4. ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .…”

Section: Numerical Exampleunclassified

“…Centrosymmetric matrices are applied in information theory, linear system theory, and numerical analysis theory [9]. e unconstrained centrosymmetric matrices' problems have been discussed [9][10][11][12][13][14], a class of unconstrained matrices' inverse eigenproblems has been obtained [15][16][17][18], and the constrained inverse eigenproblems have been discussed [19][20][21][22], but only when the eigenvalues are real or imaginary numbers. For general real matrices, the eigenvalues are not necessarily real or imaginary numbers, so when the eigenvalue is complex, it is difficult to find the constraint solution.…”

Section: Introductionmentioning

confidence: 99%

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The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem

Shen

Liu

Wang

et al. 2020

Mathematical Problems in Engineering

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In this paper, a kind of constrained inverse eigenproblem and optimal approximation problem for centrosymmetric matrices are considered. Necessary and sufficient conditions of the solvability for the constrained inverse eigenproblem of centrosymmetric matrices in real number field are derived. A general representation of the solution is presented for a solvable case. The explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.

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Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints

Hajarian¹,

Abbas²

2018

Computers & Mathematics with Applications

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On the bisymmetric nonnegative inverse eigenvalue problem

Cited by 3 publications

References 15 publications

Centrosymmetric nonnegative realization of spectra

Centrosymmetric nonnegative realization of spectra

The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem

Least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints

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