(-1,0,1)-Matrix

A -matrix is a matrix whose elements consist only of the numbers , 0, or 1. The number of distinct - matrices (counting row and column permutations, the transpose, and multiplication by as equivalent) having different row and column sums for , 4, 6, ... are 1, 4, 39, 2260, 1338614, ... (OEIS A049475). For example, the matrix is given by

To get the total number from these counts (assuming that 0 is not the missing sum, which is true for ), multiply by . In general, if an -matrix which has different column and row sums (collectively called line sums; Bodendiek and Burosch 1995), then

1. is even.

2. The number in that does not appear as a line sum is either or .

3. Of the largest line sums, half are column sums and half are row sums.

For an -matrix, the largest possible determinants (Hadamard's maximum determinant problem) are the same as for a (-1,1)-matrix, i.e., 1, 2, 4, 16, 48, 160, ... (OEIS A003433; Ehrlich 1964, Brenner and Cummings 1972) for , 2, .... The numbers of -matrices having maximum determinants are 1, 4, 240, 384, 30720, ... (OEIS A051753).

See also

Alternating Sign Matrix, C-Matrix, Integer Matrix

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References

Bodendiek, R. and Burosch, G. "Solution to the Antimagic Matrix Problem." Aufgabe 5.30 in Streifzüge durch die Kombinatorik: Aufgaben und Lösungen aus dem Schatz der Mathematik-Olympiaden. Heidelberg, Germany: Spektrum Akademischer Verlag, pp. 250-253, 1995.Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626-630, 1972.Ehrlich, H. "Determinantenabschätzungen für binäre Matrizen." Math. Z. 83, 123-132, 1964.Sloane, N. J. A. Sequences A003433/M1291, A049475, and A051753 in "The On-Line Encyclopedia of Integer Sequences."

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(-1,0,1)-Matrix

Cite this as:

Weisstein, Eric W. "(-1,0,1)-Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/-101-Matrix.html

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