Doubly structured mapping problems of the form Δx=y and Δ^*z=w

2022/08/26 by Mohit Kumar Baghel, Punit Sharma, Baghel, Mohit Kumar +1
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Matrix Theory and Algorithms #Numerical methods for differential equations #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems

paper · pdf · doi:10.48550/arxiv.2208.12429

openalex publication_date 2022/08/26 · openalex created_date 2022/08/30 · openalex updated_date 2026/07/28

Abstract

For a given class of structured matrices \mathbb S, we find necessary and sufficient conditions on vectors x,w∈ \C^n+m and y,z ∈ \Cⁿ for which there exists Δ=[Δ₁~Δ₂] with Δ₁ ∈ \mathbb S and Δ₂ ∈ \C^n,m such that Δx=y and Δ^*z=w. We also characterize the set of all such mappings Δ and provide sufficient conditions on vectors x,y,z, and w to investigate a Δ with minimal Frobenius norm. The structured classes \mathbb S we consider include (skew)-Hermitian, (skew)-symmetric, pseudo(skew)-symmetric, J-(skew)-symmetric, pseudo(skew)-Hermitian, positive (semi)definite, and dissipative matrices. These mappings are then used in computing the structured eigenvalue/eigenpair backward errors of matrix pencils arising in optimal control.

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