0,4

The entries are restricted to 0 and 1; the determinant is computed in the field of real numbers.

Suppose M = (m(i,j)) is an n X n matrix of real numbers. Let

a(n) = max det M subject to m(i,j) = 0 or 1 [this sequence],

g(n) = max det M subject to m(i,j) = -1 or 1 [A003433],

h(n) = max det M subject to m(i,j) = -1, 0 or 1 [A003433],

F(n) = max det M subject to 0 <= m(i,j) <= 1 [this sequence],

G(n) = max det M subject to -1 <= m(i,j) <= 1 [A003433].

Then a(n) = F(n), g(n) = h(n) = G(n), g(n) = 2^(n-1)*a(n-1). Thus all five problems are equivalent.

Hadamard proved that a(n) <= 2^(-n)*(n+1)^((n+1)/2), with equality if and only if a Hadamard matrix of order n+1 exists. Equivalently, g(n) <= n^(n/2), with equality if and only if a Hadamard matrix of order n exists. It is believed that a Hadamard matrix of order n exists if and only if n = 1, 2 or a multiple of 4 (see A036297).

J. Brenner, The Hadamard maximum determinant problem, Amer. Math. Monthly, 79 (1972), 626-630.

H. Ehlich, Determinantenabschaetzungen fuer binaere Matrizen, Math. Z. 83 (1964), 123-132.

H. Ehlich and K. Zeller, Binaere Matrizen, Zeit. Angew. Math. Mech., 42 (1962), 20-21.

J. Hadamard, R\'{e}solution d'une question relative aux d\'{e}terminants, Bull. des Sciences Math. 2 (1893), 240-246.

Hudelson, Matthew; Klee, Victor and Larman, David, Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem. Proceedings of the Fourth Conference of the International Linear Algebra Society (Rotterdam, 1994). Linear Algebra Appl. 241/243 (1996), 519-598.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 54.

J. Williamson, Determinants whose elements are 0 and 1, Amer. Math. Monthly 53 (1946), 427-434. Math. Rev. 8,128g.

C. Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.

W. P. Orrick, The maximal {-1, 1}-determinant of order 15.

W. P. Orrick and B. Solomon, Large determinant sign matrices of order 4k+1

W. P. Orrick and B. Solomon, The Hadamard Maximal Determinant Problem (website)

W. P. Orrick, B. Solomon, R. Dowdeswell and W. D. Smith, New lower bounds for the maximal determinant problem

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, (0, 1)-Matrix

Eric Weisstein's World of Mathematics, (-1, 0, 1)-Matrix

Index entries for sequences related to binary matrices

One of 2 ways to get determinant 9 with a 6 X 6 matrix, found by Williamson:

100110

001111

111001

010101

010011

011110

A003433(n) = 2^(n-1)*a(n-1). Cf. A013588, A036297, A051752.

Adjacent sequences: A003429 A003430 A003431 this_sequence A003433 A003434 A003435

Sequence in context: A105180 A094206 A118998 this_sequence A081938 A129500 A109204

nonn,hard,nice

njas

For a(18) through a(22) we have a(18) = 3411968?, a(19) = 19531250, a(20) = 56640625, a(21) = 195312500?, a(22) = 662671875?. See the Orrick-Solomon web site for further information.

Entry revised Jan 18 2004.