Constructive solution of the common invariant cone problem

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.

Figures (2)

• Figure 1: The sequence of cones $K_j$ generated by the Direct Algorithm for matrices from Example \ref{['ex_direct_algorithm']}. The cone $K_{\min}$ is plotted with a dashed line. The sequence $K_j$ converges to $K_{\min}$ but does not reach it.
• Figure 2: A cone generated by vectors $\left\lbrace \left[\!204\!\right], \left[\!142\!\right], \left[\!421\!\right] \right\rbrace$ (the most light one), together with its scaled versions by factors of $1/2$ and $1/4$ (the darker ones) with respect to the . center vector ${\boldsymbol{c}}$. (Picture generated using plotm from ttoolboxesttoolboxes).

Theorems & Definitions (33)

• Definition 1
• Definition 2
• Theorem 1
• Corollary 1
• Corollary 2
• Definition 3
• Definition 4
• Theorem 2
• Lemma 1
• Proposition 1