Papers › Dual-Valued Functions of Dual Matrices with Applications in Causal Emergence
Dual-Valued Functions of Dual Matrices with Applications in Causal Emergence
Tong Wei, Weiyang Ding, Yimin Wei
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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≤ p≤∞) and the unitarily invariant dual-valued Ky Fan p-k-norm (1≤ p≤∞). 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 (EI_d). Additionally, we elucidate the correlation between the 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.
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