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arXiv · 2208.12429

Doubly structured mapping problems of the form $\Delta x=y$ and $\Delta^*z=w$

Mohit Kumar Baghel·Punit Sharma
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Abstract

For a given class of structured matrices $\mathbb S$, we find necessary and sufficient conditions on vectors $x,w\in \C^{n+m}$ and $y,z \in \C^{n}$ for which there exists $\Delta=[\Delta_1~\Delta_2]$ with $\Delta_1 \in \mathbb S$ and $\Delta_2 \in \C^{n,m}$ such that $\Delta x=y$ and $\Delta^*z=w$. We also characterize the set of all such mappings $\Delta$ and provide sufficient conditions on vectors $x,y,z$, and $w$ to investigate a $\Delta$ 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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Mohit Kumar Baghel, Punit Sharma. 2022-08-26. Doubly structured mapping problems of the form $\Delta x=y$ and $\Delta^*z=w$. https://arxiv.org/abs/2208.12429

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