Hello, I am looking for some words to describe what going on here. I'm sure this is not an original thought, so I'd like to read up on more from others who have thought out this topic further.


**FORMAT**
The problem is stated in a square matrix, consider the rows and columns having identical labels (labels are shown below but are not part of the matrix):



```
  A B C D
A 0 1 2 3
B 1 0 1 2 
C 2 1 0 1
D 3 2 1 0

```

This matrix represents the distance between each row and column. In this case the dimensionality of the matrix is one, because the labels [1 2 3 4] can be plotted on the number line (say, at 1, 2, 3, 4, respectively) such that the distance between each point exactly matches the matrix.


**THE QUESTION**
What is this called? Is there an algorithm to determine the dimensionality of such a matrix? Is there any sense in thinking of a non-integer number of dimensions? Are there any other interesting questions that can be asked along this line of thought?



Dimensionality of a map -- distance
Testing whether $n$ points "need" $n-1$ dimensions (in the way you're referring to) using only their distances is done via Cayley-Menger determinants : <http://mathworld.wolfram.com/Cayley-MengerDeterminant.html>


More generally, you might want to take a look at this : <http://en.wikipedia.org/wiki/Distance_geometry>


