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Hadamard's Maximum Determinant Problem

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Hadamard's maximum determinant problem asks to find the largest possible determinant (in absolute value) for any n×n matrix whose elements are taken from some set. Hadamard (1893) proved that the determinant of any complex n×n matrix A with entries in the closed unit disk |a_(ij)|<=1 satisfies

|detA|<=n^(n/2),
(1)

with equality attained by the Vandermonde matrix of the n roots of unity (Faddeev and Sominskii 1965, p. 331; Brenner and Cummings 1972). The first few values for max|detA_n| for n=1, 2, ... are 1, 2, 3sqrt(3), 16, 25sqrt(5), 216, ..., and the squares of these are 1, 4, 27, 256, 3125, ... (OEIS A000312).

A (-1,1)-matrix attaining the bound n^(n/2) is known as a Hadamard matrix (Brenner and Cummings 1972). The same bound applies to all (-1,1)-matrices, and sharper bounds are known for orders n>2 that are 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

|detA|<=((n+1)^((n+1)/2))/(2^n)
(2)

(Faddeev and Sominskii 1965, problem 523; Brenner and Cummings 1972).

For an n×n (0,1)-matrix (i.e., a binary matrix), the largest possible determinants beta_n for n=1, 2, ... are 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477, ... (OEIS A003432). The numbers of distinct n×n binary matrices having the largest positive determinant are 1, 3, 3, 60, 3600, 529200, 75600, 195955200, 13716864000, ... (OEIS A051752).

nmatrices
1[1]
2[1 0; 0 1],[1 0; 1 1],[1 1; 0 1]
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]

For an n×n (-1,1)-matrix, the largest possible determinants alpha_n for n=1, 2, ... are 1, 2, 4, 16, 48, 160, ... (OEIS A003433; Ehlich and Zeller 1962, Ehlich 1964). The numbers of distinct n×n (-1,1)-matrices having the largest positive determinant are 1, 4, 96, 384, 30720, ... (OEIS A188895, indexed there by n-1). alpha_n is related to the largest possible (0,1)-matrix determinant beta_(n-1) by

alpha_n=2^(n-1)beta_(n-1)
(3)

for n>=2 (Williamson 1946, Brenner and Cummings 1972).

nmatrices
1[1]
2[-1 -1; 1 -1],[-1 1; -1 -1],[1 -1; 1 1],[1 1; -1 1]

For an n×n (-1,0,1)-matrix, the largest possible determinants gamma_n are the same as alpha_n (Ehlich 1964, Brenner 1972). The numbers of n×n (-1,0,1)-matrices having maximum positive determinants are 1, 4, 240, 384, 30720, ... (OEIS A051753).

For order 198, Georgiou and Stylianou (2026) attained the Ehlich-Wojtas bound by constructing a D-optimal matrix. Thus the largest absolute determinant of a (-1,1)-matrix of that order is 2^(99)·197·98^(98).

For the different entry alphabet consisting of the three cube roots of unity, Dong (2026) reports the exact order-15 maximum

max_(H in {1,omega,omega^2}^(15×15))|detH|=120932352sqrt(19),
(4)

where omega is a primitive cube root of unity. The displayed extremizing matrix has this determinant. The upper bound depends on finite exact searches and certificates. The paper determines the value, without classifying every equality case.

See also

(0,1)-Matrix, (-1,0,1)-Matrix, (-1,1)-Matrix, Determinant, Hadamard Matrix, Integer Matrix

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References

Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626-630, 1972.Cohn, J. H. E. "Determinants with Elements +/-1." J. London Math. Soc. 14, 581-588, 1963.Dong, D. "The Maximal Determinant of Order 15 over the Third Roots of Unity." 25 Oct 2026. https://arxiv.org/abs/2610.40679v1.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. Problems in Higher Algebra. San Francisco, CA: W. H. Freeman, 1965.Georgiou, S. D. and Stylianou, S. "An Exact Compression-And-Lifting Procedure and D-Optimal Designs of Orders 198 and 170." 24 Oct 2026. https://arxiv.org/abs/2610.33418.Hadamard, J. "Résolution d'une question relative aux déterminants." Bull. Sci. Math. 17, 240-246, 1893.Hall, M. Combinatorial Theory, 2nd ed. New York: Wiley, 1998.Kaplansky, I. "Never Too Late." Amer. Math. Monthly 102, 259, 1995.MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-Correcting Codes. Amsterdam, Netherlands: North-Holland, p. 54, 1978.Orrick, W. and Solomon, B. "Known Bounds on Maximal Determinants." https://web.archive.org/web/20120930030439/http://www.indiana.edu/%7Emaxdet/bounds.html.Sloane, N. J. A. Sequences A000312, A003432/M0720, A003433/M1291, A051752, A051753, and A188895 in "The On-Line Encyclopedia of Integer Sequences."Williamson, J. "Determinants Whose Elements Are 0 and 1." Amer. Math. Monthly 53, 427-434, 1946.Yang, C. H. "Some Designs for Maximal (+1,-1)-Determinant of Order n=2 (mod 4)." Math. Comput. 20, 147-148, 1966.Yang, C. H. "A Construction for Maximal (+1,-1)-Matrix of Order 54." Bull. Amer. Math. Soc. 72, 293, 1966.Yang, C. H. "On Designs of Maximal (+1,-1)-Matrices of Order n=2 (mod 4)." Math. Comput. 22, 174-180, 1968.Yang, C. H. "On Designs of Maximal (+1,-1)-Matrices of Order n=2 (mod 4) II." Math. Comput. 23, 201-205, 1969.

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Hadamard's Maximum Determinant Problem

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Weisstein, Eric W. "Hadamard's Maximum Determinant Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HadamardsMaximumDeterminantProblem.html

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