[Paper Review] On the distance from a matrix polynomial to matrix polynomials with two prescribed eigenvalues

E. Kokabifar, Ghasem Barid Loghmani+2 more|arXiv (Cornell University)|Jan 2, 2014

Matrix Theory and Algorithms14 references3 citations

TL;DR

This paper extends prior work on matrix polynomial eigenvalue perturbation by introducing lower and upper bounds for the spectral norm distance from a matrix polynomial to the set of polynomials having two prescribed distinct eigenvalues. Using a novel perturbation construction based on singular values and optimization, the method computes bounds that tightly bracket the true distance, validated through numerical examples with explicit perturbations achieving the target eigenvalues.

ABSTRACT

Consider an $n imes n$ matrix polynomial $P(λ)$. A spectral norm distance from $P(λ)$ to the set of $n imes n$ 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.

Motivation & Objective

• To generalize prior work on matrix polynomial distance to multiple eigenvalues, specifically for two distinct complex eigenvalues.
• To compute computable lower and upper bounds for the spectral norm distance from a matrix polynomial to the set of polynomials with two prescribed eigenvalues.
• To construct an explicit perturbation of the matrix polynomial that achieves the target eigenvalues while staying within the derived distance bounds.
• To validate the method through numerical examples demonstrating the tightness of the bounds and correctness of the perturbation.

Proposed method

• Introduces a spectral norm distance from a matrix polynomial $ P(\lambda) $ to the set $ \mathcal{P}_{\mu_1,\mu_2} $ of $ n\times n $ matrix polynomials having two prescribed distinct eigenvalues $ \mu_1, \mu_2 \in \mathbb{C} $.
• Derives a lower bound for the distance using singular value analysis of a structured matrix $ F[P(\mu_1,\mu_2); \gamma] $, parameterized by $ \gamma $.
• Constructs an associated perturbation $ \Delta(\lambda) $ of $ P(\lambda) $ such that the perturbed polynomial $ Q(\lambda) = P(\lambda) + \Delta(\lambda) $ has $ \mu_1 $ and $ \mu_2 $ as eigenvalues.
• Uses optimization (via MATLAB's fminsearch) to find $ \gamma_* $ that minimizes the singular value function $ s_k(F[P(\mu_1,\mu_2); \gamma]) $, enabling tight bound computation.
• Applies weighted perturbation sets $ \mathcal{B}(P, \varepsilon, w) $ with nonnegative weights $ w_j $ to control the norm of coefficient-wise perturbations.
• Validates the method by constructing explicit perturbations $ \Delta_{\gamma_*}(\lambda) $ that lie on the boundary of the perturbation set and achieve the desired eigenvalues.

Experimental results

Research questions

• RQ1 What is the spectral norm distance from a given matrix polynomial to the set of matrix polynomials that have two prescribed distinct eigenvalues?
• RQ2 How can tight lower and upper bounds for this distance be computed efficiently using singular value analysis and optimization?
• RQ3 Can an explicit perturbation of the matrix polynomial be constructed such that the perturbed polynomial has the two target eigenvalues and lies within the computed distance bounds?
• RQ4 How do the bounds behave when the optimization parameter $ \gamma_* $ is zero versus positive?
• RQ5 To what extent are the computed bounds optimal, and what conditions would yield the optimal bounds?

Key findings

• For a $ 2\times 2 $ matrix polynomial with randomly generated coefficients, the lower bound was $ \beta_{\text{low}} = 0.0376 $ and the upper bound was $ \beta_{\text{up}} = 0.2847 $, with $ \gamma_* = 1.8914 $, confirming the bounds bracket the true distance.
• The constructed perturbation $ \Delta_{1.8914}(\lambda) $ successfully yielded a matrix polynomial $ Q_{1.8914}(\lambda) $ having $ \mu_1 = 1 $ and $ \mu_2 = 2+i $ as eigenvalues, lying on the boundary of the perturbation set.
• For the case $ \gamma_* = 0 $, the method produced a perturbation $ \Delta_0 $ with norm $ \|\Delta_0\| \approx 4.1545 $, and the perturbed polynomial had eigenvalues $ \mu_1 = 5 $, $ \mu_2 = -1 $, confirming the method's robustness.
• The singular value $ s_3(F[P(1,2+i); \gamma_*]) = 4.1132 $ was minimized at $ \gamma_* = 1.8914 $, indicating optimal parameter choice for bound computation.
• Verification of Lemma 2.5 and Corollary 2.6 showed residual norms of $ 1.5612 \times 10^{-5} $ and $ 2.9886 \times 10^{-5} $, respectively, confirming theoretical consistency.
• The bounds are not necessarily optimal, but the true distance $ \mathcal{D}(P,\mu_1,\mu_2) $ is guaranteed to lie within $ [\beta_{\text{low}}, \beta_{\text{up}}] $, with the interval narrowing as $ \gamma_* $ is optimized.