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.

翻译：本文提出了一种创新的对偶延拓方法，该方法基于G\^ateaux导数，将实矩阵的实值函数推广至对偶矩阵的对偶值函数。理论上，我们给出了对偶值向量范数与矩阵范数的一般形式，这些范数保留了实域中的基本性质。特别地，我们重点研究了$p$范数$(1\!\leq\! p\!\leq\!\infty)$下的对偶值向量范数，以及酉不变的对偶值Ky Fan $p$-$k$范数$(1\!\leq\! p\!\leq\!\infty)$。随后，我们证明了该对偶值Ky Fan $p$-$k$范数与对偶矩阵前$k$个奇异值的对偶值向量$p$范数之间的等价性。在实际应用方面，我们定义了双对偶转移概率矩阵及其对应的对偶值有效信息${\rm{EI_d}}$。此外，我们阐明了${\rm{EI_d}}$、对偶值Schatten $p$范数与DTPM动力学可逆性之间的关联。通过对哑铃马尔可夫链的数值实验，我们的研究结果表明：在区间$[1,2)$内调整$p$，使得对偶值Ky Fan $p$-$k$范数的无穷小部分取得最大值时所对应的$k$值，表征了系统发生因果涌现时的最优分类数目。