So I've been playing with Euclidean Distance Matrices these days (EDMs). Super fun.
Obviously I had to list the properties that I expected an EDM would have, right at the beginning of the process, and I didn't realize something back than, that I've stumbled upon right now.
So let $D$ be an $N \times N$ Euclidean Distance Matrix, which, as the name implies, is the matrix in which each entry $D_{ij}$ represents the Euclidean distance between two points ${\vec p}_i$ and ${\vec p}_j$ in $\mathbb{R}^m$.
EDIT: User Thomas Andrews pointed out to me in the comments that this definition isn't the common definition of an Euclidean Distance Matrix. In the usual definition, the elements $D_{ij}$ are the squares of the distances. I wasn't aware of that when I first stumbled upon this small proof so I will keep here the original definitions that led me to the question. Beware.
According to what I know about Euclidean distances, I have decided, as a starting point, that the elements $D_{ij}$ of a well-behaved tax-paying EDM should have these properties:
- Real: $D_{ii} \in \mathbb{R} \forall i,j$
- Traceless: $D_{ii} = 0$
- Symmetric: $D_{ij} = D_{ji}$
- Nonegative $D_{ij} \geq 0 \forall i \neq j$
- Metric: $D_{ij} \leq D_{ik} + D_{kj} \forall i,j,k$
The names of the properties are arbitrary, just so we can have a hook on which to hang these definitions. I'm not a trained mathematician, I'm a chemist, there are probably names for these things that I'm not using here and for that I apologize.
So I started playing with these definitions in order to test whether I had all my bases covered. At first I wanted to test whether the property $(5)$ really held for any triplets $(i,j,k)$. So I did what any sensible investigator here would do and set $i=j$ in order to test it.
In that situation, if the matrix is Metric then we have that $$\ \tag{1} D_{ii} \leq D_{ik} + D_{ki} $$ But if $D$ is a also Symmetric, then $D_{ik}=D_{ki}$, which means eq. $(1)$ becomes: $$ \tag{2} D_{ii} \leq 2 D_{ik} $$ If $D$ is Traceless, however, it is true that $D_{ii}=0$ so the inequality in eq $(2)$ becomes $$ \tag{3} 0 \leq D_{ik} $$ If you multiply eq $(3)$ by $-1$ it becomes $D_{ik} \geq 0$, which is just a statement that $D$ is Nonegative.
So here is my question:
If $D$ is Real, Traceless, Symmetric and Metric, dos that imply that it is Nonegative, or is there some kind of restriction on the triplet of indices in the triangle inequality? For instance, that they should be distinct.
If that is the case then an EDM has to satisfy only the following properties:
- Real: $D_{ii} \in \mathbb{R} \forall i,j$
- Traceless: $D_{ii} = 0$
- Symmetric: $D_{ij} = D_{ji}$
- Metric: $D_{ij} \leq D_{ik} + D_{kj} \forall i,j,k$