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- W1968627178 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 λ i , singular values σ i , and inverse B = ( b ij ). We are interested in the four closely related problems of finding λ ( n ) = min A , i | λ i |, σ ( n ) = min A , i σ i , χ ( n ) = max A , i , j | b ij |, and μ ( n ) = max A Σ ij b 2 ij . Then A is an anti-Hadamard matrix if it attains μ( n ). We show that λ( n ), σ( n ) are between (2n) −1 ( n 4 ) −n 2 and c √ n (2.274) − n , where c is a constant, c (2.274) n ⩽χ(n)⩽2( n 4 ) n 2 , and c (5.172) n ⩽μ(n)⩽4n 2 ( n 4 ) n . 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." @default.
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- W1968627178 date "1984-11-01" @default.
- W1968627178 modified "2023-09-26" @default.
- W1968627178 title "Anti-Hadamard matrices" @default.
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- W1968627178 doi "https://doi.org/10.1016/0024-3795(84)90090-9" @default.
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