65.125 HadamardMat

HadamardMat( n )

HadamardMat returns a Hadamard matrix of order n. This is an n by n matrix with the property that the matrix multiplied by its transpose returns n times the identity matrix. This is only possible for <n>=1, <n>=2 or in cases where n is a multiple of 4. If the matrix does not exist or is not known, HadamardMat returns an error. A large number of construction methods is known to create these matrices for different orders. HadamardMat makes use of two construction methods (among which the Sylvester construction (see SylvesterMat)). These methods cover most of the possible Hadamard matrices, although some special algorithms have not been implemented yet. The following orders less than 100 do not have an implementation for a Hadamard matrix in GUAVA: 28, 36, 52, 76, 92.

    gap> C := HadamardMat(8);; PrintArray(C);
    [ [   1,   1,   1,   1,   1,   1,   1,   1 ],
      [   1,  -1,   1,  -1,   1,  -1,   1,  -1 ],
      [   1,   1,  -1,  -1,   1,   1,  -1,  -1 ],
      [   1,  -1,  -1,   1,   1,  -1,  -1,   1 ],
      [   1,   1,   1,   1,  -1,  -1,  -1,  -1 ],
      [   1,  -1,   1,  -1,  -1,   1,  -1,   1 ],
      [   1,   1,  -1,  -1,  -1,  -1,   1,   1 ],
      [   1,  -1,  -1,   1,  -1,   1,   1,  -1 ] ]
    gap> C * TransposedMat(C) = 8 * IdentityMat( 8, 8 );
    true 

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