Properties
It follows from the definition that a Hadamard matrix H of order n satisfies
where In is the n × n identity matrix and HT is the transpose of H. Consequently the determinant of H equals ±nn/2.
Suppose that M is a complex matrix of order n, whose entries are bounded by |Mij| ≤1, for each i, j between 1 and n. Then Hadamard's determinant bound states that
Equality in this bound is attained for a real matrix M if and only if M is a Hadamard matrix.
The order of a Hadamard matrix must be 1, 2, or a multiple of 4.
Read more about this topic: Hadamard Matrix
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—Ralph Waldo Emerson (1803–1882)
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