Abstract
This paper gives a rigorous and greatly simplified proof of Guttman's theorem for the least upper-bound dimensionality of arbitrary real symmetric matricesS, where the points embedded in a real Euclidean space subtend distances which are strictly monotone with the off-diagonal elements ofS. A comparable and more easily proven theorem for the vector model is also introduced. At mostn-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.
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References
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This research in nonmetric methods is supported in part by a grant from the National Science Foundation (GS-929 & -2850).
The very helpful comments and encouragement of Louis Guttman and J. Douglas Carroll are greatly appreciated. Finally, to that unknown, but not unsung, reviewer who helped in the clarification of the argument, I express my thanks.
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Lingoes, J.C. Some boundary conditions for a monotone analysis of symmetric matrices. Psychometrika 36, 195–203 (1971). https://doi.org/10.1007/BF02291398
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DOI: https://doi.org/10.1007/BF02291398