The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

Xie, Dongxiu; Hu, Xi-Yan; Zhang, Lei

doi:10.1002/nla.285

Cited by 25 publications

(14 citation statements)

References 8 publications

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“…In [5,[16][17][18] it is shown that S(Z, ) is nonempty if and only (Z, ) satisfies certain conditions involving Moore-Penrose inverses or singular value decompositions of matrices related to (Z, ). However, no method is given for choosing pairs (Z, ) that satisfy the conditions and, because of the restricted spectral structures of the matrices in question, it is not likely that a randomly chosen (Z, ) will be admissible.…”

Section: Problem 1 (Inverse Eigenproblem)mentioning

confidence: 98%

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Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry

Trench

2004

Linear Algebra and its Applications

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Let R ∈ C n×n be a nontrivial involution; i.e., R = R −1 / = ±I . We say that A ∈ C n×n is R-symmetric (R-skew symmetric) if RAR = A (RAR = −A). Let S be one of the following subsets of C n×n : (i) R-symmetric matrices; (ii) Hermitian R-symmetric matrices; (iii) Rskew symmetric matrices; (iv) Hermitian R-skew symmetric matrices. Let Z ∈ C n×m with rank(Z) = m and = diag(λ 1 , . . . , λ m ).The inverse eigenproblem consists of finding (Z, ) such that the set S(Z, ) = {A ∈ S|AZ = Z } is nonempty, and to find the general form of A ∈ S(Z, ). In all cases we use the special spectral properties of S to essentially characterize the set of admissible pairs (Z, ), and the special structure of the members of S to obtain the general solution of the inverse eigenproblem.Given an arbitrary B ∈ S, the approximation problem consists of finding the unique matrix A B ∈ S( , Z) that best approximates B in the Frobenius norm.It is not necessary to assume that R = R * in connection with the inverse eigenproblem for R-symmetric or R-skew symmetric matrices. However, we impose this additional assumption in connection with the inverse eigenproblem for Hermitian R-symmetric or R-skew symmetric matrices, and in connection with the approximation problem for (i)-(iv).

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Section: Problem 1 (Inverse Eigenproblem)mentioning

confidence: 98%

“…Recently the following problems have received attention [5,[16][17][18]. These references describe applications in which such problems arise.…”

Section: Introductionmentioning

confidence: 98%

Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry

Trench

2004

Linear Algebra and its Applications

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“…On this topic of PDIEPs, the earlier study can be found for real symmetric Toeplitz matrices [1,13] and Jacobi matrices in [1,14], and some of the recent works can be found for anti-symmetric matrices in [15], anti-persymmetric matrices in [16], centrosymmetric matrices in [17], symmetric anti-bidiagonal matrices in [18], K-symmetric matrices in [19], and K-centrohermitian matrices in [12]. This is by far not a complete list, see [2] for a recent review, a number of applications and an extensive list of references.…”

Section: Introductionmentioning

confidence: 97%

The reconstruction of an hermitian toeplitz matrix with prescribed eigenpairs

2010

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This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, the authors first consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related reconstruction problem. The authors show that the dimension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, and the solution of the reconstruction problem of an hermitian Toeplitz matrix with two given eigenpairs is unique.

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“…They also occur in a remarkable variety of applications [2,[5][6][7][8] and have been studied for different classes of structured matrices. We refer the reader to [9][10][11][12] and references therein. For example, Zhou et al [10] and Zhang et al [12] considered the problems for the case of centro-symmetric matrices and Hermitian-generalized Hamiltonian matrices, respectively.…”

Section: Introductionmentioning

confidence: 99%

An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices

Xie¹,

Huang²,

Zhang³

2007

Numer Algor

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Based on the theory of inverse eigenvalue problem, a correction of an approximate model is discussed, which can be formulated as N X = X , where X and are given identified modal and eigenvalues matrices, respectively. The solvability conditions for a symmetric skew-Hamiltonian matrix N are established and an explicit expression of the solutions is derived. For any estimated matrix C of the analytical model, the best approximation matrixN to minimize the Frobenius norm of C − N is provided and some numerical results are presented. A perturbation analysis of the solutionN is also performed, which has scarcely appeared in existing literatures.

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The solvability conditions for inverse eigenproblem of symmetric and anti‐persymmetric matrices and its approximation

Cited by 25 publications

References 8 publications

Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry

Inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry

The reconstruction of an hermitian toeplitz matrix with prescribed eigenpairs

An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices

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