QUESTION. What are the differences between a "dissimilarity matrix" and a "distance matrix"? Do they share the same properties, or not? In case, can you show an example?
If there is any difference among these two matrices, to me, that difference is not clear when I read these two references:
- Encyclopedia of Distances, by Michel Marie Deza and Elena Deza
- Matrix Algebra: Theory, Computations, and Applications in Statistics, by James E. Gentle
EDITED 1. "Dissimilarity matrix" from Encyclopedia of Distances, by Michel Marie Deza and Elena Deza
EDITED 2. "Dissimilarity matrix" from Matrix Algebra: Theory, Computations, and Applications in Statistics, by James E. Gentle
I doubt many readers will track down the definitions in those references, but I can give you my take on the terms. Neither strikes me has having a universally agreed-upon definition. "Distance matrix" is close, however: an $n \times n$ matrix whose entries $d_{ij}$ form a metric on $\{1,2,\dots,n\}$. In particular, the $d_{ij}$ must obey the triangle inequality: $d_{ij} + d_{jk} \ge d_{ik}$. However, if an author allows an off-diagonal $d_{ij} = 0$ in their distance matrix, that seems reasonable: calling this a "semi-distance matrix" seems too pedantic.
On the other hand, the term "dissimilarity matrix" is much vaguer. If I encountered this term, I'd expect the matrix to be zero on the diagonal and positive otherwise. I'd expect the matrix to be symmetric, but I wouldn't be too surprised if it weren't. I wouldn't expect the triangle inequality to hold. I might expect there to be a similarity matrix to be defined with entries in $[0,1]$ and a dissimilarity matrix to be one minus this.
In short, some terms in mathematics, such as "group" or "real number" have universally agreed-upon definitions. "Distance matrix" is close to having one, but "dissimilarity matrix" is not. The latter is more of a description of how to think about the elements of a matrix than a definition of a type of matrix. Of course, authors are always free to define what they mean by "dissimilarity matrix" or any other term.