On the real angular stability radius for sectorial matrices

Sei Zhen Khong; Xin Mao; Li Qiu; Axel Ringh

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arXiv · 2608.30843

On the real angular stability radius for sectorial matrices

Sei Zhen Khong·Xin Mao·Li Qiu·Axel Ringh

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Abstract

The real stability radius problem asks for the smallest real perturbation that makes a given matrix lose rank, and it is a structured counterpart of the corresponding complex stability radius problem. In this paper, we consider an angular version of this question for sectorial matrices, where the size of a perturbation is measured by its largest phase instead of its largest singular value. More precisely, for a sectorial matrix $M$, we investigate the minimum required largest phase of a real and sectorial perturbation $\Delta$ for which $I+M\Delta$ loses rank. We show that for some matrices, the complex and the real angular stability radius are the same, but that they are in general different, such as when $M$ is diagonal or complex symmetric. For these cases, we derive a formula for the real angular stability radius, and this formula depends on the two largest phases of $M$. Moreover, in these cases we show that the result for $n \times n$ matrices can be reduced to that for $2 \times 2$ matrices by compressing an arbitrary real destabilizing perturbation to the real two-dimensional subspace spanned by the real and imaginary parts of a destabilizing vector.

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Sei Zhen Khong, Xin Mao, Li Qiu, Axel Ringh. 2026-08-31. On the real angular stability radius for sectorial matrices. https://arxiv.org/abs/2608.30843

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