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When <math>P_i</math> is a solitary heat vector without uncertainty, the equal sign of this equation holds.  This occurs when <math>P_i</math> is a deterministic one-hot vector. Therefore, when all <math>P_i</math> are one-hot vectors, we have:

When <math>P_i</math> is a one-hot vector without uncertainty, the equal sign of this equation holds.  This occurs when <math>P_i</math> is a deterministic one-hot vector. Therefore, when all <math>P_i</math> are one-hot vectors, we have:

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Established. On the other hand, there is also an inequality that holds for the entropy of the average vector of all row vectors in P,

On the other hand, there is also an inequality that holds for the entropy of the average vector of all row vectors in P,

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The equal sign holds when the condition <math>\bar{P}=\frac{1}{N}\cdot \mathbb{1}</math> is satisfied.

The equal sign holds when the condition <math>\bar{P}=\frac{1}{N}\cdot \mathbb{1}</math> is satisfied, where [math]\mathbb{1}[/math] represents the vector with all elements being 1.

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When the equal sign of these two equations holds simultaneously, that is, when <math>P_i</math> is a solitary heat vector and <math>\bar{P}</math> is uniformly distributed (at this time, it is necessary to require [math]P_i[/math] to be perpendicular to each other, that is, P is a [[Permutation Matrix]]) — EI reaches its maximum value of:

When the equal sign of these two equations holds simultaneously, that is, when <math>P_i</math> is a one-hot vector and <math>\bar{P}</math> is uniformly distributed (at this time, it is necessary to require [math]P_i[/math] to be perpendicular to each other, that is, P is a [[Permutation Matrix]]) — EI reaches its maximum value of:

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