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- Hadamards_maximal_determinant_problem abstract "Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2n−1 times the maximal determinant of a {0,1} matrix of size n−1. The problem was posed by Hadamard in the 1893 paper in which he presented his famous determinant bound and remains unsolved for matrices of general size. Hadamard's bound implies that {1, −1}-matrices of size n have determinant at most nn/2. Hadamard observed that a construction of Sylvesterproduces examples of matrices that attain the bound when n is a power of 2, and produced examples of his own of sizes 12 and 20. He also showed that the bound is only attainable when n is equal to 1, 2, or a multiple of 4. Additional examples were later constructed by Scarpis and Paley and subsequently by many other authors. Such matrices are now known as Hadamard matrices. They have received intensive study.Matrix sizes n for which n ≡ 1, 2, or 3 (mod 4) have received less attention. The earliest results are due to Barba, who tightened Hadamard's bound for n odd, and Williamson, who found the largest determinants for n=3, 5, 6, and 7. Some important results include tighter bounds, due to Barba, Ehlich, and Wojtas, for n ≡ 1, 2, or 3 (mod 4), which, however, are known not to be always attainable, a few infinite sequences of matrices attaining the bounds for n ≡ 1 or 2 (mod 4), a number of matrices attaining the bounds for specific n ≡ 1 or 2 (mod 4), a number of matrices not attaining the bounds for specific n ≡ 1 or 3 (mod 4), but that have been proved by exhaustive computation to have maximal determinant.The design of experiments in statistics makes use of {1, −1} matrices X (not necessarily square) for which the information matrix XTX has maximal determinant. (The notation XT denotes the transpose of X.) Such matrices are known as D-optimal designs. If X is a square matrix, it is known as a saturated D-optimal design.".
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- Hadamards_maximal_determinant_problem wikiPageRevisionID "671921917".
- Hadamards_maximal_determinant_problem wikiPageWikiLink Block-diagonal_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Block_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Category:Design_theory.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Category:Matrices.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Category:Unsolved_problems_in_mathematics.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Cofactor_expansion.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Design_of_experiments.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Determinant.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Fisher_information.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Gram_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Gramian_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Hadamard_matrices.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Hadamard_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Hadamards_inequality.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Hasse-Minkowski_theorem.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Hasse–Minkowski_theorem.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Identity_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Information_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Jacques_Hadamard.
- Hadamards_maximal_determinant_problem wikiPageWikiLink James_Joseph_Sylvester.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Laplace_expansion.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Matrix_(mathematics).
- Hadamards_maximal_determinant_problem wikiPageWikiLink Normal_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Odd_number.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Optimal_design.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Orthogonal.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Orthogonality.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Parity_(mathematics).
- Hadamards_maximal_determinant_problem wikiPageWikiLink Positive-definite_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Projective_plane.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Square_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Statistics.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Symmetric_matrix.
- Hadamards_maximal_determinant_problem wikiPageWikiLink Transpose.
- Hadamards_maximal_determinant_problem wikiPageWikiLinkText "Hadamard's maximal determinant problem".
- Hadamards_maximal_determinant_problem wikiPageWikiLinkText "maximum determinant".
- Hadamards_maximal_determinant_problem hasPhotoCollection Hadamards_maximal_determinant_problem.
- Hadamards_maximal_determinant_problem wikiPageUsesTemplate Template:Reflist.
- Hadamards_maximal_determinant_problem subject Category:Design_theory.
- Hadamards_maximal_determinant_problem subject Category:Matrices.
- Hadamards_maximal_determinant_problem subject Category:Unsolved_problems_in_mathematics.
- Hadamards_maximal_determinant_problem comment "Hadamard's maximal determinant problem, named after Jacques Hadamard, asks for the largest determinant of a matrix with elements equal to 1 or −1. The analogous question for matrices with elements equal to 0 or 1 is equivalent since, as will be shown below, the maximal determinant of a {1,−1} matrix of size n is 2n−1 times the maximal determinant of a {0,1} matrix of size n−1.".
- Hadamards_maximal_determinant_problem label "Hadamard's maximal determinant problem".
- Hadamards_maximal_determinant_problem sameAs m.0g9wdns.
- Hadamards_maximal_determinant_problem sameAs Q5637579.
- Hadamards_maximal_determinant_problem sameAs Q5637579.
- Hadamards_maximal_determinant_problem wasDerivedFrom Hadamards_maximal_determinant_problemoldid=671921917.
- Hadamards_maximal_determinant_problem isPrimaryTopicOf Hadamards_maximal_determinant_problem.