Hadamard's maximum determinant problem asks to find the largest possible determinant (in absolute value) for any 
matrix whose elements are taken from some set. Hadamard
(1893) proved that the determinant of any complex 
matrix 
with entries in the closed unit disk 
satisfies
 |
(1)
|
with equality attained by the Vandermonde matrix of the 
roots of unity (Faddeev and
Sominskii 1965, p. 331; Brenner and Cummings 1972). The first few values for
for 
,
2, ... are 1, 2, 
, 16, 
, 216, ..., and the squares of these are 1, 4, 27,
256, 3125, ... (OEIS A000312).
A (-1,1)-matrix having a maximal determinant is known as a Hadamard matrix (Brenner and Cummings 1972).
The same bound of 
applies to such matrices, and sharper bounds are known
when the size of the matrix is not a multiple of 4. A summary of what is known about
such bounds is given by Orrick and Solomon.
For a (0,1)-matrix, Hadamard's bound can be improved to
 |
(2)
|
(Faddeev and Sominskii 1965, problem 523; Brenner and Cummings 1972).
For an 
(0,1)-matrix (i.e., a
binary matrix), the largest possible determinants 
for 
, 2, ... are 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645,
9477, ... (OEIS A003432). The numbers of distinct
binary matrices having the largest possible determinant are 1, 3, 3, 60, 3600, 529200,
75600, 195955200, 13716864000, ... (OEIS A051752).
 | matrices |
| 1 | ![[1]](/images/equations/HadamardsMaximumDeterminantProblem/Inline16.svg) |
| 2 | ![[1 0; 0 1],[1 0; 1 1],[1 1; 0 1]](/images/equations/HadamardsMaximumDeterminantProblem/Inline17.svg) |
| 3 | ![[0 1 1; 1 0 1; 1 1 0],[1 0 1; 1 1 0; 0 1 1],[1 1 0; 0 1 1; 1 0 1]](/images/equations/HadamardsMaximumDeterminantProblem/Inline18.svg) |
For an 
(-1,1)-matrix, the largest
possible determinants 
for 
, 2, ... are 1, 2, 4, 16, 48, 160, ... (OEIS A003433;
Ehlich and Zeller 1962, Ehlich 1964). The numbers of distinct 
-matrices having the largest possible determinant are
1, 4, 96, 384, 30720, ... (OEIS A188895). 
is related to the largest possible 
-matrix determinant 
by
 |
(3)
|
(Williamson 1946, Brenner and Cummings 1972).
 | matrices |
| 1 | ![[1]](/images/equations/HadamardsMaximumDeterminantProblem/Inline28.svg) |
| 2 | ![[-1 -1; 1 -1],[-1 1; -1 -1],[1 -1; 1 1],[1 1; -1 1]](/images/equations/HadamardsMaximumDeterminantProblem/Inline29.svg) |
For an 
(-1,0,1)-matrix, the
largest possible determinants 
are the same as 
(Ehlich 1964, Brenner 1972). The numbers of 
-matrices having maximum determinants are 1, 4, 240,
384, 30720, ... (OEIS A051753).
." J. London Math. Soc. 14, 581-588,
1963.Ehlich, H. "Determinantenabschätzungen für binäre
Matrizen." Math. Z. 83, 123-132, 1964.Ehlich, H.
and Zeller, K. "Binäre Matrizen." Z. angew. Math. Mechanik 42,
T20-21, 1962.Faddeev, D. K. and Sominskii, I. S. 
-Determinant of Order 
." Math. Comput. 20, 147-148,
1966.Yang, C. H. "A Construction for Maximal 
-Matrix of Order 54." Bull. Amer. Math. Soc. 72,
293, 1966.Yang, C. H. "On Designs of Maximal 
-Matrices of Order 
." Math. Comput. 22, 174-180,
1968.Yang, C. H. "On Designs of Maximal 
-Matrices of Order 
II." Math. Comput. 23, 201-205,
1969.