On the nearest stable 2��2 matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas

Abstract

In this paper, we study the continuous-time nearest stable matrix problem: given a \begin{document}$ 2\times 2 $\end{document} real matrix \begin{document}$ A $\end{document} , minimize the Frobenius norm of \begin{document}$ A-X $\end{document} , where \begin{document}$ X $\end{document} is a stable matrix. We provide an explicit formula for the global minimizer \begin{document}$ X_* $\end{document} . The uniqueness of the minimizer is also studied.