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arXiv · 2505.01229

Constructive solution of the common invariant cone problem

Thomas Mejstrik·Vladimiar Yu. Protasov
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Abstract

Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be algorithmically undecidable for general sets of matrices. We show that it can nevertheless be efficiently solved in practice. An algorithm that for a given finite set of matrices, either finds a common invariant cone or proves its non-existence is presented. Numerical results demonstrate that it works for a vast majority of matrix sets. The structure and properties of the minimal and maximal invariant cones are analyzed. Applications to dynamical systems and combinatorics are considered.

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Thomas Mejstrik, Vladimiar Yu. Protasov. 2025-05-02. Constructive solution of the common invariant cone problem. https://arxiv.org/abs/2505.01229

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