From: dmosher@blackhole.nyx.net (David Mosher) Subject: Re: Weighing matrices? Date: Tue, 23 May 2000 10:26:07 GMT Newsgroups: sci.math.research Summary: [missing] Mon 22 May 2000, in article , Robin Chapman wrote, > I have an n-by-n skewsymmetric matrix M with the property that all > off-diagonal entries are +1 or -1 and M^2 = -(n-1)I. > > I also have the vague memory that I have seen the term "weighing matrix" > used for this (or a similar) type of matrix. Can anyone confirm this, or > better, give an accessible reference to an article/book which uses this > term? [9; p437] defines a *weighing matrix* as a square {-1,0,1}-matrix, W, such that WW^t = kI for some k (a non-negative integer). ("W^t" denotes the transpose of W.) [7; p173], citing the work of Belevitch ([2], [3]), defines a *conference matrix of order n* to be an nxn matrix, C, with 0 diagonal and 1 or -1 in all off-diagonal positions such that CC^t = (n-1)I. In [6], a matrix satisfying the definition of a conference matrix which is also either symmetric or skew-symmetric is called a *C-matrix*. Recall that a *Hadamard matrix of order n* is an nxn {-1,1}-matrix, H, such that HH^t = nI. A necessary condition that a Hadamard matrix of order n exist is that n is 1, 2, or a multiple of 4. It is conjectured that this is also sufficient. Many partial results and special cases are known; see [1] and [9]. A Hadamard matrix H such that H-I is skew-symmetric is said to be *of skew type* or *skew Hadamard* ([4; p61]). For the matrix M quoted above (a skew- symmetric conference matrix, or, a skew C-matrix), M+I is skew Hadamard. It is also conjectured that skew Hadamard matrices exist for all multiple-of-4 orders. Among the partial results known, notable is Paley's construction using quadratic characters on finite fields, which yields the existence of skew Hadamard matrices of orders q+1 for prime-powers q congruent to 3 (mod 4); see [8], [4; 55--58], or [7; 175--176]. For other results, see [1], [5], [6], and [9]. References: [1] Agaian, S.S.: Hadamard matrices and their applications. Lecture Notes in Math. #1168, Springer-Verlag, 1985. (MR 87k:05038) [2] Belevitch, V.: Theory of 2n-terminal networks with applications to conference telephony. Electrical Communication 27 (1950), 231--244. [3] Belevitch, V.: Conference networks and Hadamard matrices. Ann. Soc. Sci. Bruxelles S\'er. I 82 (1968), 13--32. (MR 39 #85) [4] Beth, Thomas; Jungnickel, Dieter; Lenz, Hanfried: Design theory. Vol. I. Second edition. Encyclopedia of Mathematics and its Applications, 69. Cambridge University Press, 1999. (CMP 1 729 456) [5] Delsarte, P.; Goethals, J.-M.; Seidel, J.J.: Orthogonal matrices with zero diagonal, II. Canad. J. Math. 23 (1971), 816--832. (MR 48 #3765) [6] Goethals, J.-M.; Seidel, J.J.: Orthogonal matrices with zero diagonal. Canad. J. Math. 19 (1967), 1001--1010. (MR 36 #5013) [7] van Lint, J.H.; Wilson, R.M.: A course in combinatorics. Cambridge University Press, 1992. (MR 94g:05003) [8] Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12 (1933), 311--320. (Zbl 007.10004) [9] Seberry, Jennifer; Yamada, Mieko: Hadamard matrices, sequences, and block designs. In: Contemporary design theory: A collection of surveys, 431--560. Dinitz, Jeffrey H.; Stinson, Douglas R. (Eds). Wiley-Interscience Series in Discrete Math. & Optimization, Wiley, 1992. (MR 94c:05001) -- David Mosher (The ultimate effect of Hawking radiation should be applied to the address.) ============================================================================== From: gmg@maths.may.ie (Gary McGuire) Subject: Weighing matrices? Date: 23 May 2000 12:30:00 -0400 Newsgroups: sci.math.research A weighing matrix, according to The CRC Handbook of Combinatorial Designs, by Charles J. Colbourn and Jeffrey H. Dinitz, (p.496) is any n-by-n matrix W with 0, +1, -1 entries satisfying W W^t = mI. In your case m=n-1, and such matrices are called conference matrices. See also J. van Lint and R.M. Wilson, A Course in Combinatorics, (check the index) for conference matrices. -Gary McGuire