Dual-Valued Functions of Dual Matrices with Applications in Causal Emergence

Dual continuation, an innovative insight into extending the real-valued functions of real matrices to the dual-valued functions of dual matrices with a foundation of the G\^ateaux derivative, is proposed. Theoretically, the general forms of dual-valued vector and matrix norms, the remaining properties in the real field, are provided. In particular, we focus on the dual-valued vector $p$-norm $(1\!\leq\! p\!\leq\!\infty)$ and the unitarily invariant dual-valued Ky Fan $p$-$k$-norm $(1\!\leq\! p\!\leq\!\infty)$. The equivalence between the dual-valued Ky Fan $p$-$k$-norm and the dual-valued vector $p$-norm of the first $k$ singular values of the dual matrix is then demonstrated. Practically, we define the dual transitional probability matrix (DTPM), as well as its dual-valued effective information (${\rm{EI_d}}$). Additionally, we elucidate the correlation between the ${\rm{EI_d}}$, the dual-valued Schatten $p$-norm, and the dynamical reversibility of a DTPM. Through numerical experiments on a dumbbell Markov chain, our findings indicate that the value of $k$, corresponding to the maximum value of the infinitesimal part of the dual-valued Ky Fan $p$-$k$-norm by adjusting $p$ in the interval $[1,2)$, characterizes the optimal classification number of the system for the occurrence of the causal emergence.