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In mathematics, a conference matrix (also called a C-matrix) is a square matrix ''C'' with 0 on the diagonal and +1 and −1 off the diagonal, such that ''C''T''C'' is a multiple of the identity matrix ''I''. Thus, if the matrix has order ''n'', ''C''T''C'' = (''n''−1)''I''. Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal.〔Malcolm Greig, Harri Haanpää, and Petteri Kaski, Journal of Combinatorial Theory, Series A, vol. 113, no. 4, 2006, pp 703-711, 〕〔Harald Gropp, More on orbital matrices, Electronic Notes in Discrete Mathematics, vol. 17, 2004, pp 179-183, 〕 Conference matrices first arose in connection with a problem in telephony.〔Belevitch, pp. 231-244.〕 They were first described by Vitold Belevitch who also gave them their name. Belevitch was interested in constructing ideal telephone conference networks from ideal transformers and discovered that such networks were represented by conference matrices, hence the name.〔Colbourn and Dinitz, (2007), p.19 van Lint and Wilson, (2001), p.98 Stinson, (2004), p.200〕 Other applications are in statistics, and another is in elliptic geometry.〔van Lint, J.H., and Seidel, J.J. (1966), Equilateral point sets in elliptic geometry. ''Indagationes Mathematicae'', vol. 28, pp. 335-348.〕 For ''n'' > 1, there are two kinds of conference matrix. Let us normalize ''C'' by, first (if the more general definition is used), rearranging the rows so that all the zeros are on the diagonal, and then negating any row or column whose first entry is negative. (These operations do not change whether a matrix is a conference matrix.) Thus, a normalized conference matrix has all 1's in its first row and column, except for a 0 in the top left corner, and is 0 on the diagonal. Let ''S'' be the matrix that remains when the first row and column of ''C'' are removed. Then either ''n'' is evenly even (a multiple of 4), and ''S'' is antisymmetric (as is the normalized ''C'' if its first row is negated), or ''n'' is oddly even (congruent to 2 modulo 4) and ''S'' is symmetric (as is the normalized ''C''). ==Symmetric conference matrices== If ''C'' is a symmetric conference matrix of order ''n'' > 1, then not only must ''n'' be congruent to 2 (mod 4) but also ''n'' − 1 must be a sum of two square integers;〔Belevitch, p.240〕 there is a clever proof by elementary matrix theory in van Lint and Seidel.〔 ''n'' will always be the sum of two squares if ''n'' − 1 is a prime power.〔Stinson, p.78〕 Given a symmetric conference matrix, the matrix ''S'' can be viewed as the Seidel adjacency matrix of a graph. The graph has ''n'' − 1 vertices, corresponding to the rows and columns of ''S'', and two vertices are adjacent if the corresponding entry in ''S'' is negative. This graph is strongly regular of the type called (after the matrix) a conference graph. The existence of conference matrices of orders ''n'' allowed by the above restrictions is known only for some values of ''n''. For instance, if ''n'' = ''q'' + 1 where ''q'' is a prime power congruent to 1 (mod 4), then the Paley graphs provide examples of symmetric conference matrices of order ''n'', by taking ''S'' to be the Seidel matrix of the Paley graph. The first few possible orders of a symmetric conference matrix are ''n'' = 2, 6, 10, 14, 18, (not 22, since 21 is not a sum of two squares), 26, 30, (not 34 since 33 is not a sum of two squares), 38, 42, 46, 50, 54, (not 58), 62 ; for every one of these, it is known that a symmetric conference matrix of that order exists. Order 66 seems to be an open problem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Conference matrix」の詳細全文を読む スポンサード リンク
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