Nearest structured matrix having an eigenvalue with prescribed lower bounds of algebraic and geometric multiplicities
Abstract
We study the problem of perturbing a given matrix X belonging to a subspace S of linearly structured real matrices of order n to its nearest counterpart Y=X+D, where D belongs to S and Y possesses an eigenvalue with prescribed lower bounds on its algebraic multiplicity and geometric multiplicity. The proposed framework takes care of both cases where the target eigenvalue is known and where it is treated as an unknown decision variable to be determined jointly with the structured perturbation. We establish necessary and sufficient conditions for the existence of such matrices by expressing the feasibility constraints through Jordan-chain relations in structural coordinates. This allows us to formulate the problem as a nonlinear constrained nested optimization that minimizes the Frobenius norm of the perturbation. To solve this, we implemented a two-level strategy, utilizing MATLAB's fmincon for continuous inner optimization via a multi-start Sequential Quadratic Programming approach, while the outer level solves a discrete optimization problem. Comprehensive numerical experiments on various classes of structured matrices, including comparisons with existing methods, demonstrate the effectiveness and accuracy of the proposed approach.