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Formal definition of protein sequence similarity

I have a very basic question: I'm looking to implement myself the calculation of similarity (not identity!) between two aligned sequences, but unfortunately, I've found the answers online to be too vague to be implemented as code, to contain dead links, or to simply say "you can get it with program/library X".

My question simply is: given two aligned protein sequences and an amino acid substitution matrix (e.g. BLOSUM62), how is the similarity between the two sequences calculated?

sequence similarity alignment

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