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 , for each i, j between 1 and n, then

(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

- http://en.wikipedia.org/wiki/Hadamard_matrix
- http://www.research.att.com/~njas/hadamard/
- Hadamard matrices and their applications, by K. J. Horadam
- 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)

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

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