# Hadamard’s maximal determinant problem

This blog entry is to remind or introduce to people the fascinating problem called the Hadamard maximal determinant problem: What is the maximal possible determinant of a matrix M whose entries are of absolute value at most 1? Hadamard proved that if M is a complex matrix of order n, whose entries are bounded by $|M_{ij}| \leq 1$, for each i, j between 1 and n, then
$|det(M)| \leq n^{n/2}$ (equality is attained, so this is best possible for such matrices).

If instead the entries of the matrix are +1 or -1 and the size of the matrix is nxn where n is a multiple of 4, then the problem of the maximal determinant presumably boils down to the well-known search for Hadamard matrices. This is discussed in many books, papers and website but in particular, I refer to

1. http://en.wikipedia.org/wiki/Hadamard_matrix

2. http://www.research.att.com/~njas/hadamard/
3. Hadamard matrices and their applications, by K. J. Horadam
4. http://www.uow.edu.au/~jennie/hadamard.html

What I think is fascinating is the entries of the matrix are only assumed to be real and not of size 4kx4k. In this case, the maximal value of the determinant is less clear. The results are complicated and depend in a fascinating way on the congruence class of n mod 4. Please see the excellent webpages (maintained by Will Orrick and Bruce Solomon)

1. http://www.indiana.edu/~maxdet/, and in particular,
2. http://www.indiana.edu/~maxdet/bounds.html

In particular, the case of an nxn matrix with n=4k+3 seems to be open.