Anti-Hadamard matrices |
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Authors: | RL Graham NJA Sloane |
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Institution: | Mathematics and Statistics Research Center Bell Laboratories Murray Hill, New Jersey 07974 USA |
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Abstract: | An anti-Hadamard matrix may be loosely defined as a real (0, 1) matrix which is invertible, but only just. Let A be an invertible (0, 1) matrix with eigenvalues λi, singular values σi, and inverse B = (bij). We are interested in the four closely related problems of finding λ(n) = minA, i|λi|, σ(n) = minA, iσi, χ(n) = maxA, i, j |bij|, and μ(n) = maxAΣijb2ij. Then A is an anti-Hadamard matrix if it attains μ(n). We show that λ(n), σ(n) are between and c√n (2.274)?n, where c is a constant, , and . We also consider these problems when A is restricted to be a Toeplitz, triangular, circulant, or (+1, ?1) matrix. Besides the obvious application—to finding the most ill-conditioned (0, 1) matrices—there are connections with weighing designs, number theory, and geometry. |
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