Long-Term Behavior of Cross-Dimensional Linear Dynamical Systems
Kuize Zhang, Karl Henrik Johansson · 2018
Let M and V denote the sets of finite-dimensional matrices and finite-dimensional column vectors, respectively. Based on the semitensor product and the vector addition, M and V both form a monoid, where V is commutative. In addition, based on an equivalence relation ↔ on V, the induced quotient space V/↔ forms a vector space. In this paper, we give a basis for the vector space V/↔, showing that V/↔ is of countably infinite dimension. In addition, we give an explicit characterization for how the dimension of a vector in V changes caused by the repetitive actions of a matrix in M on the vector, and characterize the generalized inverse behavior of the repetitive actions.