A similarity canonical form for max-plus matrices and its eigenproblem
Haicheng Zhang, Xiyan Zhu
TL;DR
This work characterizes when a max-plus matrix is similar to a pseudo-diagonal canonical form, providing a necessary and sufficient condition for pseudo-diagonalizability and an efficient $O(n^2)$ test. It then derives closed-form powers for pseudo-diagonal matrices, analyzes the invariance of optimal-node and separable matrices under similarity, and characterizes their eigenstructure within the pseudo-diagonalizable class. The eigenproblem for general pseudo-diagonalizable matrices is resolved by reducing to the canonical form, yielding explicit expressions for eigenvalues and eigenspaces in terms of the maximal diagonal entry and the corresponding columns. Collectively, the results offer practical tools for spectral analysis and long-term dynamics in max-plus linear systems via similarity to a tractable canonical form.
Abstract
We provide a necessary and sufficient condition for matrices in the max-plus algebra to be pseudo-diagonalizable, calculate the powers of pseudo-diagonal matrices and prove the invariance of optimal-node matrices and separable matrices under similarity. As an application, we determine the eigenvalues and eigenspaces of pseudo-diagonalizable matrices.