Hadamard's Maximum Determinant Problem

Author

Eric W. Weisstein
April 19, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/HadamardsMaximumDeterminantProblem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HadamardsMaximumDeterminantProblem.html.

©2011 Wolfram Research, Inc. except for portions noted otherwise

(-1,0,1)

max=Table[Max[Det/@Matrices[i,{-1,0,1}]],{i,3}]

max={1,2,4,16,48,160,576,4096};

Table[Count[Det/@Matrices[i,{-1,0,1}],max[[i]]],{i,3}]//Timing

Table[Count[Det/@Matrices[i,{-1,0,1}],max[[i]]],{i,4}]//Timing

MaxMemoryUsed[]/10^9. GB

Table[MatrixForm/@Select[Matrices[i,{-1,0,1}],Det[#]==max[[i]]&],{i,3}]//TableForm

(-1,1)

α={1,2,4,16,48,160,576,4096,14336}

β={1,1,2,3,5,9,32,56,144,320,1458,3645,9477};

Table[β[[n-1]]2^(n-1),{n,2,10}]

MatrixForm/@Matrices[2,{-1,1}]

max=Table[Max[Det/@Matrices[i,{-1,1}]],{i,4}]

Table[Count[Det/@Matrices[i,{-1,1}],max[[i]]],{i,4}]//Timing

MaxMemoryUsed[]/10^9.GB

max={1,2,4,16,48,160,576,4096,14336,73728};

Table[Count[Det/@Matrices[i,{-1,1}],max[[i]]],{i,5}]//Timing

MaxMemoryUsed[]/10^9.GB

Table[MatrixForm/@Select[Matrices[i,{-1,1}],Det[#]==max[[i]]&],{i,4}]//TableForm

(0,1)

Complex matrix bound

Floor[Table[n^(n/2),{n,10}]]

Real matrix bound

Floor[Table[(n+1)^((n+1)/2)/2^n,{n,10}]]

Floor[Table[Sqrt[(2n-1)(n-1)^(n-1)],{n,10}]]

β={1,1,2,3,5,9,32,56,144,320,1458,3645,9477}

Max[Det/@Matrices[#]]&/@Range[4]//Timing

Table[Count[Det/@Matrices[i],β[[i]]],{i,4}]//Timing

MaxMemoryUsed[]/10^9.GB

Table[Count[Det/@Matrices[i],β[[i]]],{i,5}]//Timing

MaxMemoryUsed[]/10^9.GB

Table[Count[Det/@Matrices[i],β[[i]]],{i,6}]//Timing

MaxMemoryUsed[]/10^9.GB

Table[MatrixForm/@Select[Matrices[i],Det[#]==β[[i]]&],{i,4}]//TableForm