Nearest matrix with prescribed eigenvalues and its applications
Provides an accessible algorithm for a known theoretical result, benefiting practitioners in numerical linear algebra.
The paper presents a practical computational method for finding the minimum spectral norm perturbation that adds prescribed eigenvalues to a matrix, extending prior theoretical work. Applications are discussed.
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.