The Idempotents of 5th-Order Fractal-Tropical Matrices and Their Applications

To address the modeling and noise-handling challenges of fractal systems, this paper integrates fractal properties into the theory of tropical matrix idempotents, proposing the concept of 5th-order fractal-tropical matrix idempotents. First, based on the fractal modification of tropical algebra, the operational rules of 5th-order fractal-tropical matrices are established, including fractal-tropical addition and multiplication, which adapt to the non-integer dimensional characteristics of fractal systems. Second, drawing on the structural analysis of tropical matrix idempotents, the necessary and sufficient conditions for 5th-order fractal-tropical matrix idempotents (satisfying [Formula: see text], where [Formula: see text] denotes the fractal dimension) are derived, and their structures are classified into three types when [Formula: see text]: full non-zero diagonal, partial non-zero diagonal and full zero diagonal. Finally, applications in fractal image denoising and fractal optimal control verify that fractal-tropical matrix idempotents outperform traditional tropical matrix idempotents in preserving fractal features and improving processing accuracy. This study enriches the theoretical system of tropical matrix idempotents and provides a new algebraic tool for fractal system analysis.