page_title: Conference matrix
section_title: Telephone conference circuits
hierarchical_section_title: Conference matrix / Telephone conference circuits
context_page_description: In mathematics, a conference matrix is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CᵀC is a multiple of the identity matrix I. Thus, if the matrix has order n, CᵀC = 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.
  Conference matrices first arose in connection with a problem in telephony. 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. Other applications are in statistics, and another is in elliptic geometry.
  For n > 1, there are two kinds of conference matrix. Let us normalize C by, first, rearranging the rows so that all the zeros are on the diagonal, and then negating any row or column whose first entry is negative.
  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.
context_section_description: Belevitch obtained complete solutions for conference matrices for all values of n up to 38 and provided circuits for some of the smaller matrices.  An ideal conference network is one where the loss of signal is entirely due to the signal being split between multiple conference subscriber ports.  That is, there are no dissipation losses within the network.  The network must contain ideal transformers only and no resistances.  An n-port ideal conference network exists if and only if there exists a conference matrix of order n.  For instance, a 3-port conference network can be constructed with the well-known hybrid transformer circuit used for 2-wire to 4-wire conversion in telephone handsets and line repeaters.  However, there is no order 3 conference matrix and this circuit does not produce an ideal conference network.  A resistance is needed for matching which dissipates signal, or else signal is lost through mismatch.
  As mentioned above, a necessary condition for a conference matrix to exist is that n−1 must be the sum of two squares.  Where there is more than one possible sum of two squares for n−1 there will exist multiple essentially different solutions for the corresponding conference network.  This situation occurs at n of 26 and 66.  The networks are particularly simple when n−1 is a perfect square (n = 2, 10, 26, …).
caption: The trivial 2-port conference network