Some Boundary Conditions for a Monotone Analysis of Symmetric Matrices

This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matrices S , where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements of S . A comparable and more easily proven theorem for the vector model is also introduced. At most n -2 dimensions are required to reproduce the order information for both the distance and vector models and this is true for any choice of real indices, whether they define a metric space or not. If ties exist in the matrices to be analyzed, then greatest lower bounds are specifiable when degenerate solutions are to be avoided. These theorems have relevance to current developments in nonmetric techniques for the monotone analysis of data matrices.