On the Inverse Symmetric Quadratic Eigenvalue Problem

@article{Lancaster2014OnTI,
  title={On the Inverse Symmetric Quadratic Eigenvalue Problem},
  author={Peter Lancaster and Ion Zaballa},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2014},
  volume={35},
  pages={254-278}
}
The detailed spectral structure of symmetric, algebraic, quadratic eigenvalue problems has been developed recently. In this paper we take advantage of these canonical forms to provide a detailed analysis of inverse problems of the following form: construct the coefficient matrices from the spectral data including the classical eigenvalue/eigenvector data and sign characteristics for the real eigenvalues. An orthogonality condition dependent on these signs plays a vital role in this construction… 
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References

SHOWING 1-10 OF 27 REFERENCES
The Quadratic Eigenvalue Problem
We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the
An orthogonality property for real symmetric matrix polynomials with application to the inverse problem
An orthogonality property common to a broad class of real symmetric matrix polynomials is developed generalizing earlier results concerning polynomials of second degree. This property is obtained
Symmetric tridiagonal inverse quadratic eigenvalue problems with partial eigendata
This paper concerns the inverse problem of constructing the n-by-n real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(λ) ≔ λ2I + λC + K (where I is the identity matrix)
DIAGONALIZABLE QUADRATIC EIGENVALUE PROBLEMS
On the Sign Characteristics of Selfadjoint Matrix Polynomials
An important role is played in the spectral analysis of selfadjoint matrix polynomials by the so-called “sign characteristics” associated with real eigenvalues. In this paper the ordering of the real
A Review of Canonical Forms for Selfadjoint Matrix Polynomials
In the theory of n×n matrix polynomials, the notions of “standard pairs and triples”, and the special cases of “Jordan pairs and triples” play an important role. There are interesting differences in
On Inverse Quadratic Eigenvalue Problems with Partially Prescribed Eigenstructure
TLDR
It is shown via construction that the inverse problem is solvable for any k, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided $k \leq n$.
Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem
TLDR
Using various matrix decompositions, a general solution to the inverse quadratic eigenvalue problem of constructing real symmetric matrices is constructed and several particular solutions with additional eigeninformation or special properties are constructed.
An Inverse Eigenvalue Problem for the Symmetric Tridiagonal Quadratic Pencil with Application to Damped Oscillatory Systems
A method is presented which constructs an n by n tridiagonal, symmetric, quadratic pencil which has its $2n$ eigenvalues and the $2n - 2$ of its $n - 1$-dimensional leading principal subpencil
Inverse Spectral Problems for Semisimple Damped Vibrating Systems
TLDR
Admissible sets of data concerning systems of eigenvalues and eigenvectors are examined, and procedures for generating associated (isospectral) families of systems are developed.
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