NAFeb 18, 2015
Nearest matrix with prescribed eigenvalues and its applicationsEsmaeil Kokabifar, Ghasem Barid Loghmani, S. M. Karbassi
Consider $n \times n$ matrix $A$ and a set $Λ$ consisting of $k \le n$ prescribed complex numbers. Lippert (2010) in a challenging article, studied geometrically the spectral norm distance from $A$ to the set $Λ$ and constructed a perturbation matrix $Δ$ with minimum spectral norm such that $A+Δ$ had $Λ$ in its spectrum. This paper presents an easy practical computational method for constructing the optimal perturbation $Δ$ by extending necessary definitions and lemmas of previous works. Also, some conceivable applications of this issue are provided.
NAMay 23, 2015
On the distance from a matrix polynomial to matrix polynomials with $k$ prescribed distinct eigenvaluesE. Kokabifar, G. B. Loghmani, P. J. Psarrakos et al.
Consider an $n\times n$ matrix polynomial $P(λ)$ and a set $Σ$ consisting of $k \le n$ distinct complex numbers. In this paper, a (weighted) spectral norm distance from $P(λ)$ to the matrix polynomials whose spectra include the specified set $Σ$, is defined and studied. An upper and a lower bounds for this distance are obtained, and an optimal perturbation of $P(λ)$ associated to the upper bound is constructed. Numerical examples are given to illustrate the efficiency of the proposed bounds.
NANov 14, 2014
On the distance from a matrix polynomial to matrix polynomials with two prescribed eigenvaluesEsmaeil Kokabifar, G. B. Loghmani, A. M. Nazari et al.
Consider an $n \times n$ matrix polynomial $P(λ)$. A spectral norm distance from $P(λ)$ to the set of $n \times n$ matrix polynomials that have a given scalar $μ\in\mathbb{C}$ as a multiple eigenvalue was introduced and obtained by Papathanasiou and Psarrakos. They computed lower and upper bounds for this distance, constructing an associated perturbation of $P(λ)$. In this paper, we extend this result to the case of two given distinct complex numbers $μ_{1}$ and $μ_{2}$. First, we compute a lower bound for the spectral norm distance from $P(λ)$ to the set of matrix polynomials that have $μ_1,μ_2$ as two eigenvalues. Then we construct an associated perturbation of $P(λ)$, such that the perturbed matrix polynomial has two given scalars $μ_1$ and $μ_2$ in its spectrum. Finally, we derive an upper bound for the distance by the constructed perturbation of $P(λ)$. Numerical examples are provided to illustrate the validity of the method.