Purpose: Orthogonal matrices with many applications introduced by J. J. Sylvester have been become famous: Hadamard
matrices, symmetric conference matrices, and weighing matrices are the best known of these matrices with entries from
the unit disk. The goal of this note is to develop a theory of such orthogonal matrices based on preliminary research results.
Methods: Extreme solutions (using the determinant) have been established by minimization of the maximum of the absolute
values of the elements of the matrices followed by their subsequent classification. Results: We show that if S is a core
of a symmetric conference weighing matrix, then there exists a three-level orthogonal matrix, X. We apply this result to the
three-level matrices given by Paley using Legendre symbols to give a new infinite family of Cretan orthogonal matrices.
An algorithmic optimization procedure is known which raises the value of the determinant. Our example is for Cretan matrices
upto, say four decimal places (but could be made more). Practical relevance: The over-riding aim is to seek Cretan matrices
as they have many applications in image processing (compression, masking) to statisticians undertaking medical or agricultural
research, and to obtain lossless circuits for telecommunications conference networking. Web addresses are given for
other illustrations and other matrices with similar properties. Algorithms to construct Cretan matrices have been implemented
in developing software of the research program-complex.