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[[Erik Hoel's causal emergence theory]] needs to specify a coarse-graining strategy in advance. Rosas' information decomposition theory does not completely solve this problem. Therefore, Zhang et al.<ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> further proposed the [[causal emergence theory based on singular value decomposition]].

[[Erik Hoel's causal emergence theory]] needs to specify a coarse-graining strategy in advance. Rosas' information decomposition theory does not completely solve this problem. Therefore, Zhang et al.<ref name=":2">Zhang J, Tao R, Yuan B. Dynamical Reversibility and A New Theory of Causal Emergence. arXiv preprint arXiv:2402.15054. 2024 Feb 23.</ref> further proposed the [[causal emergence theory based on singular value decomposition]].

=====Singular value decomposition of Markov chains=====

=====Singular value decomposition of Markov chain=====

Given the [[Markov transition matrix]] <math>P</math> of a system, we can perform [[singular value decomposition]] on it to obtain two orthogonal and normalized matrices <math>U</math> and <math>V</math>, and a diagonal matrix <math>\Sigma</math>: <math>P = U\Sigma V^T</math>. Here, [math]\Sigma = diag(\sigma_1,\sigma_2,\cdots,\sigma_N)[/math], where [math]\sigma_1\geq\sigma_2\geq\cdots\sigma_N[/math] are the singular values of <math>P</math> and are arranged in descending order. <math>N</math> is the number of states of <math>P</math>.

Given the [[Markov transition matrix]] <math>P</math> of a system, we can perform [[singular value decomposition]] on it to obtain two orthogonal and normalized matrices <math>U</math> and <math>V</math>, and a diagonal matrix <math>\Sigma</math>: <math>P = U\Sigma V^T</math>. Here, [math]\Sigma = diag(\sigma_1,\sigma_2,\cdots,\sigma_N)[/math], where [math]\sigma_1\geq\sigma_2\geq\cdots\sigma_N[/math] are the singular values of <math>P</math> and are arranged in descending order. <math>N</math> is the number of states of <math>P</math>.