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In more general situations, if the row vectors of P resemble a matrix formed by independent one-hot vectors, P becomes less degenerate. On the other hand, if the row vectors are identical and close to a one-hot vector, P becomes more degenerate.
In more general situations, if the row vectors of P resemble a matrix formed by independent one-hot vectors, P becomes less degenerate. On the other hand, if the row vectors are identical and close to a one-hot vector, P becomes more degenerate.
===Example===
===Examples===
Below, we examine the determinism and degeneracy of three Markov chains.
Below, we examine the determinism and degeneracy of three Markov chains.
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The first transition probability matrix is a [[Permutation Matrix]], which is invertible. It has the highest determinism and no degeneracy, leading to the maximum EI. The second matrix has the first three states transitioning to one another with equal probability (1/3), resulting in the lowest determinism but non-degeneracy, with EI being 0.81. The third matrix is deterministic but since three of the states transition to the first state, it's impossible to infer from state 1 which previous state led to it. Therefore, it has high degeneracy, and its EI is also 0.81, the same as the second.
The first transition probability matrix is a [[Permutation Matrix]], which is invertible. It has the highest determinism and no degeneracy, leading to the maximum EI. The second matrix has the first three states transitioning to one another with equal probability (1/3), resulting in the lowest determinism but zero degeneracy, with EI being 0.81. The third matrix is deterministic but since three of the states transition to the first state, it's impossible to infer from state 1 which previous state led to it. Therefore, it has high degeneracy, and its EI is also 0.81, the same as the second.
===Normalized Determinism and Degeneracy===
===Normalized Determinism and Degeneracy===