About: Hadamard's maximal determinant problem

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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.

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rdfs:label	Hadamard's maximal determinant problem (en)
rdfs: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. 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. (en)
sameAs	Hadamard's maximal determinant problem Hadamard's maximal determinant problem
dbp:wikiPageUsesTemplate	dbt:Reflist
Subject	Combinatorial design Unsolved problems in mathematics Matrices
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem?oldid=1034258364&ns=0
Wikipage page ID	30556829 (xsd:integer)
page length (characters) of wiki page	17328 (xsd:nonNegativeInteger)
Wikipage revision ID	1034258364 (xsd:integer)
Link from a Wikipage to another Wikipage	Normal matrix Projective plane Block matrix Statistics Combinatorial design Unsolved problems in mathematics Jacques Hadamard James Joseph Sylvester Matrices Definite matrix Square matrix Optimal design Orthogonality Parity (mathematics) Fisher information Gram matrix Unit matrix Symmetric matrix Design of experiments Determinant Hadamard's inequality Hadamard matrix Hasse–Minkowski theorem Transpose Laplace expansion Matrix (mathematics)
has 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. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem
is Wikipage redirect of	Hadamard's determinant problem Hadamard's maximum determinant problem
is Link from a Wikipage to another Wikipage of	List of things named after Jacques Hadamard List of unsolved problems in mathematics 1000 (number) 144 (number) 1458 (number) 300 (number) 56 (number) Hadamard matrix Hadamard's determinant problem Hadamard's maximum determinant problem
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Hadamard's_maximal_determinant_problem

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