# 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}
}
• Published 6 March 2014
• Mathematics
• SIAM J. Matrix Anal. Appl.
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…
14 Citations
On the Sign Characteristics of Selfadjoint Matrix Polynomials
• Mathematics
• 2013
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
Matrix Polynomials with Completely Prescribed Eigenstructure
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2015
The proof developed for the existence of a matrix polynomial when its degree, its finite and infinite elementary divisors, and its left and right minimal indices are prescribed is constructive and solves a very general inverse problem for Matrix polynomials with prescribed complete eigenstructure.
The Polynomial Eigenvalue Problem is Well Conditioned for Random Inputs
• Mathematics, Computer Science
SIAM J. Matrix Anal. Appl.
• 2019
We compute the exact value of the squared condition number for the polynomial eigenvalue problem, when the input matrices have entries coming from the standard complex Gaussian distribution, showing
Normalization of Eigenvectors and Certain Properties of Parameter Matrices Associated with The Inverse Problem for Vibrating Systems
• Mathematics
• 2016
Solutions of the equation of motion of an n-dimensional vibrating system $$M\ddot{q} + D\dot{q} + Kq = 0$$ can be found by solving the quadratic eigenvalue problem \(L(\lambda )x:=\lambda ^{2}Mx
Solution of the linearly structured partial polynomial inverse eigenvalue problem
• Mathematics, Computer Science
J. Comput. Appl. Math.
• 2022
Quadratic realizability of palindromic matrix polynomials: the real case
• Mathematics
Linear and Multilinear Algebra
• 2022
Let L = ( L 1 , L 2 ) be a list consisting of structural data for a matrix polynomial; here L 1 is a sublist consisting of powers of irreducible ( monic ) scalar polynomials over the ﬁeld R , and L 2
Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2019
The structural data of any rational matrix $R(\lambda)$, i.e., the structural indices of its poles and zeros together with the minimal indices of its left and right nullspaces, is known to satisfy ...
Balanced truncation model reduction for symmetric second order systems - A passivity-based approach
• Mathematics, Computer Science
SIAM J. Matrix Anal. Appl.
• 2021
We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the
Inverse Spectral Problems for Linked Vibrating Systems and Structured Matrix Polynomials
• Mathematics
• 2017
We show that for a given set of $nk$ distinct real numbers $\Lambda$, and $k$ graphs on $n$ nodes, $G_0, G_1,\ldots,G_{k-1}$, there are real symmetric $n\times n$ matrices $A_s$, $s=0,1,\ldots, k$

## References

SHOWING 1-10 OF 27 REFERENCES
• Mathematics
SIAM Rev.
• 2001
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
• Mathematics
• 2013
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)
On the Sign Characteristics of Selfadjoint Matrix Polynomials
• Mathematics
• 2013
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
• Mathematics
• 2012
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
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2004
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
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2006
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
• Mathematics
SIAM J. Appl. Math.
• 1996
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
Admissible sets of data concerning systems of eigenvalues and eigenvectors are examined, and procedures for generating associated (isospectral) families of systems are developed.