I'm reading a paper and it mentioned the following: "one obtains this self-consistent matrix formulation:" and then they mentioned a formulation. But I couldn't understand the meaning of a self-consistent matrix.
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$\begingroup$Indeed, it's the "matrix formulation" that's self-consistent, i.e., the formula used to define the similarity index $S$ discussed in the paper.
The reason this is noted is that $S$ is defined in terms of $S$ itself, so it's not immediate that it's even well-defined.
The manipulations in (14) show that it can be equivalently defined non-recursively, and thus is indeed well-defined.
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