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Characteristic polynomials - factorisable or not

"An invariant of [the distance matrix D = (d(i, j))], independent of the ordering of the points, is the characteristic polynomial of D." Care to define this concept, and give an example or two? (In particular, one that factorises and another that doesn't.)

- YahyaA October 28, 2015, 06:45:27 AM UTC-0700

The distance matrix completely characterises a finite metric space

We've heard that "A finite metric space is completely characterized by its distance matrix D = (d(i, j))". Again, a concrete example or two would make this concept clearer.

- YahyaA October 28, 2015, 06:41:06 AM UTC-0700

Concrete examples help understand abstract concepts

It would be useful to follow the Definition with one or two specific examples, showing the actual distance values for several notes of a specific periodic scale.

- YahyaA October 28, 2015, 06:35:07 AM UTC-0700