Conference matrix

The conference matrix , also known as the C matrix , is a square matrix which has the value 0 on the main diagonal and only includes the values +1 and −1 in all other positions, so that it represents a multiple of the identity matrix. That is, the C matrix of the order of the equation: ${\ displaystyle \ mathbf {C}}$ ${\ displaystyle \ mathbf {C} ^ {T} \ mathbf {C}}$ ${\ displaystyle \ mathbf {I}}$ ${\ displaystyle n}$

{\ displaystyle \ mathbf {C} ^ {T} \ mathbf {C} = (n-1) \ mathbf {I} \,}

enough. In addition, there is another generalized definition which only requires that the element 0 must appear once in every row and column and that the position of the 0 is not restricted to the main diagonal.

Conference matrices are used, among other things, in the area of the design of conference circuits in the area of telephone networks and their circuit-theoretical description and were first formulated by Vitold Belevitch , who also coined the term. The conference matrix serves as a criterion for determining whether an ideal passive conference circuit consisting only of ideal transformers theoretically and without losses in the coupling network can in principle exist according to adapters such as terminating resistors to adapt different line impedances for a certain number of conference participants. Further applications are in the field of statistics and elliptical geometry .

Classification

There are two different types of order in the conference matrix . For subdivision, the matrix is normalized, which does not change the properties of a conference matrix. To do this, all lines that start with a are negated . Then all columns are negated with one at the top . The normalized conference matrix formed in this way only has the value 1 in the first column and in the first row, with the exception of the top left position of the matrix with the value 0. Let be a matrix formed from this, in which the first column and first row of the normalized conference matrix are removed. Then there is either a multiple of 4 and a skew-symmetric matrix , in this case the underlying conference matrix is called skew-symmetric, or is congruent to 2 modulo 4 and a symmetric matrix . In the latter case, the underlying conference matrix is called symmetrical. ${\ displaystyle n> 1}$ ${\ displaystyle \ mathbf {C}}$ ${\ displaystyle \ mathbf {C}}$ ${\ displaystyle -1}$ ${\ displaystyle -1}$ ${\ displaystyle (0,0)}$ ${\ displaystyle \ mathbf {S}}$ ${\ displaystyle n}$ ${\ displaystyle \ mathbf {S}}$ ${\ displaystyle n}$ ${\ displaystyle \ mathbf {S}}$

Symmetrical conference matrix

For the existence of a conference matrix with order which is symmetrical in the following , the sum of two squares must be. The evidence can be found in. If, as a special case, a prime number power , this condition is always fulfilled, since then equals the sum of two squares. ${\ displaystyle \ mathbf {C}}$ ${\ displaystyle n> 1}$ ${\ displaystyle n-1}$ ${\ displaystyle n-1}$ ${\ displaystyle n}$

The existence of symmetric conference matrices is only known for a few cases of . The known orders are in the sequence A000952 in OEIS : ${\ displaystyle n}$

{\ displaystyle n = \ {2,6,10,14,18,26,30,38,42,46,50,54,62 \} \,}

example

The normalized conference matrix of order 6 is given as: ${\ displaystyle \ mathbf {C} _ {6}}$

{\ displaystyle \ mathbf {C} _ {6} = {\ begin {pmatrix} 0 & + 1 & + 1 & + 1 & + 1 & + 1 \\ + 1 & 0 & + 1 & -1 & -1 & + 1 \\ + 1 & + 1 & 0 & + 1 & -1 & -1 \\ + 1 & -1 & + 1 & 0 & + 1 & -1 \\ + 1 & -1 & -1 & + 1 & 0 & + 1 \\ + 1 & + 1 & -1 & -1 & + 1 & 0 \ end {pmatrix}}}

All other conference matrices of order 6 can be formed by inverting the sign of any rows and columns.

Ideal conference calls in the telephony area

Trivial "conference call" with two participants

Vitold Belevitch was able to provide the solutions for all existing symmetrical conference matrices up to order 38 and for some of the smaller orders it was able to specify specific electrical circuits for realizing ideal conference circuits. In this context, an ideal conference call is an electrical coupling network that has no losses, only uses ideal transformers for transmission and distributes the signal of one participant evenly to all other participants.

The difficulty in the implementation is that a subscriber connection, consisting of the two connections ( ma, mb ), with m the subscriber number, has an identical and the same line impedance for all participants and at the same time an ideal conference call must not have any terminating resistances, otherwise these terminating resistors in the coupling network would represent a certain loss of signal and there would be no ideal conference call. Likewise, a mismatch with unequal line wave resistances leads to a loss of signal energy in the coupling network.

An ideal conference call for participants only exists if the symmetrical conference matrix of the order exists. For example, there is no solution for an ideal conference call with 3 participants - but conference calls with 3 participants can also be implemented, for example with the aid of the hybrid connection . However, additional terminating resistors are necessary and their signal loss does not result in an ideal conference call. ${\ displaystyle n}$ ${\ displaystyle n}$

Circuit implementation of ideal conference calls
For 6 participants
For 10 participants

In those cases in which there is more than a sum of two squares for for a certain order , there are just as many different, but equivalent and functionally identical and ideal conference circuits. This is the case with order 26. The circuits can be built from transformers with a simple 1: 1 turn ratio of the turns if is a perfect square. This is the case. ${\ displaystyle n}$ ${\ displaystyle (n-1)}$ ${\ displaystyle (n-1)}$ ${\ displaystyle n = \ {2,10,26 \}}$

Individual evidence

↑ Harald Gropp: More on orbital matrices . In: Electronic Notes in Discrete Mathematics . tape 17 , 2004, ISSN 1571-0653 , p. 179-183 , doi : 10.1016 / j.endm.2004.03.036 .
↑ ^a ^b ^c ^d Vitold Belevitch : Theorem of 2 n -terminal networks with application to conference telephony . In: Electrical Communication . tape 26 , 1950, ISSN 1242-0565 , p. 231-244 ( online ).
↑ Damaraju Raghavarao: Some Optimum Weighing Designs. In: Annals of Mathematical Statistics . Vol. 30, No. 2, 1959, pp. 295-303, online , doi : 10.1214 / aoms / 1177706253 .
↑ ^a ^b J. H. van Lint, JJ Seidel: Equilateral point sets in elliptic geometry. In: Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series A: Mathematical Sciences. Vol. 69, No. 3, 1966, ISSN 0023-3358 , pp. 335-348, online (PDF; 638 kB) .
^ Douglas R. Stinson: Combinatorial Designs. Constructions and Analysis . Springer, New York NY et al. 2004, ISBN 0-387-95487-2 .

Web links

C-Matrix , Wolfram MathWorld, 2012, engl.

[Gropp-1] Harald Gropp: More on orbital matrices . In: Electronic Notes in Discrete Mathematics . tape 17 , 2004, ISSN 1571-0653 , p. 179-183 , doi : 10.1016 / j.endm.2004.03.036 .

[Belev-2] Vitold Belevitch : Theorem of 2 n -terminal networks with application to conference telephony . In: Electrical Communication . tape 26 , 1950, ISSN 1242-0565 , p. 231-244 ( online ).

[Raghavarao-3] Damaraju Raghavarao: Some Optimum Weighing Designs. In: Annals of Mathematical Statistics . Vol. 30, No. 2, 1959, pp. 295-303, online , doi : 10.1214 / aoms / 1177706253 .

[vL-4] J. H. van Lint, JJ Seidel: Equilateral point sets in elliptic geometry. In: Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen. Series A: Mathematical Sciences. Vol. 69, No. 3, 1966, ISSN 0023-3358 , pp. 335-348, online (PDF; 638 kB) .

[Stinson-5] Douglas R. Stinson: Combinatorial Designs. Constructions and Analysis . Springer, New York NY et al. 2004, ISBN 0-387-95487-2 .