User:Will Orrick/Hadamard's maximal determinant problem

The difficulties both in finding maximal-determinant matrices and in proving maximality depend on the congruence class of the matrix size n modulo 4. When n≡0 (mod 4), the upper bound on the determinant is conjectured to be always attainable. In other words, it is conjectured that a Hadamard matrix exists of every size n≡0 (mod 4), Proof of maximality is immediate when a Hadamard matrix is found. Many constructions of Hadamard matrices are known, including constructions that build larger Hadamard matrices out of smaller ones. As a result, Hadamard matrices are known of a great many sizes. When n≡1, 2, or 3 (mod 4), the known upper bounds are not always attainable. In the cases n≡1 or 2 mod (4), constructions are known that attain the upper bounds infinitely often; many additional constructions are known that attain the upper bound for specific values of n. When n≡3 (mod 4), the upper bound is not yet known to be attained for any value of n except for n=3. The next smallest size for which the upper bound might be attained is n=511.

Proving maximality in cases where the upper bound is not attained is a difficult computational problem; again, the nature of the difficulty is sensitive the the congruence class modulo 4. The case n≡1 (mod 4) appear to be the easiest, and computer proofs of maximality have been obtained in sizes n=9, 17, 21, and 37. The case n≡3 (mod 4) is more difficult; proofs have been obtained in sizes n=7, 11, 15, and 19. No proofs of maximality are known in cases when the upper bound is not attained and n≡2 (mod 4). Plausible candidates for the maximal determinant are known for n=22, 34, 58, 70, 78, 94, and 106.

Real matrices with bounded elements
The maximal determinant of an n×n real matrix whose elements are bounded in magnitude by 1 is the same as the maximal determinant of an n×n {1,−1}-matrix. This is a consequence of expansion by minors; the determinant is maximized only when the (i,j)th element has the same sign as the associated cofactor and the largest possible magnitude. This implies that the (i,j)th element can only be ±1 unless that associated cofactor is 0, in which case the (i,j)th element can arbitrarily be set to 1 or −1 without affecting the determinant.

Complex matrices with bounded elements
Hadamard's bound can always be attained for n×n complex matrices whose elements are restricted to lie within the unit disk. Matrices that attain the bound are known as complex Hadamard matrices, one example of which is the n×n Vandermonde matrix for the nth roots of unity.

Upper bounds on the maximal determinant
If the rows of an n×n matrix R are regarded as n-dimensional vectors, one can form the matrix of inner products of these vectors, that is, the matrix G whose (i,j)th element is the dot product of row i and row j of R.  Equivalently, G=RRT where RT is the transpose of R.  This matrix is the Gram matrix of the rows. Likewise, we may form the Gram matrix of columns, G' = RTR. A matrix is normal if G = G', but this need not be the case for maximal determinant matrices.

Examples
1: Syl, Had 2: Syl, Had 3: Wil 4: Syl, Had 5: Wil 6: Wil 7: Wil 8: Syl, Had 9: Ehl-Zel 10: Ehl-Zel 11: Gal-Kie (Ehl) 12: Had 13: Rha 14: Ehl 15: Smi-Coh, Orr 16: Syl, Had 17: Moy-Kou 18: Ehl

Lower bounds
Clements and Lindström..., de Launey and Levin (SIAM J. Disc. Math. 2009)

Related problems
Although the maximal determinant problem is already quite difficult, even more ambitious questions can be asked. One such questions is the determinant spectrum problem, first studied by Metropolis.