Hadamard's maximal determinant problem

Unsolved problems in mathematics | Combinatorial design | Matrices

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. (Wikipedia).

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Hadamard's maximal determinant problem

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