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Just published an interactive article about a magical algorithm known as the Burrows-Wheeler Transform, which powers sequence alignment tools like bowtie and bwa: sandbox.bio/concepts/bwt It's also notoriously unintuitive so I'm hoping this article helps you build that intuition.

It’s not obvious to me what happens in this step. You say it “sorts the matrix” but I don’t see that at all. The two strings swap places and one completely changes to an order that is very much not sorted. What am I missing?

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Ah, by sorting the matrix, I meant sorting the set of 9 strings defined by each row of the matrix, not sorting the 2 strings defined by each column of the matrix.