where HT is the transpose of H, I is the identity matrix, and λ is a scalar quantity. They are usually written in “normalized” form, i.e. the rows and columns have been signed so that the top row and left column consist of +1 elements only. Hadamard matrices exist only for order m = 1, 2, or 4r for some r. It is known that they exist for all orders m = 2s. It is conjectured, but not known that they exist for all orders m = 4r.
The rows of any Hadamard matrix form an orthonormal basis, from which property follows many of their applications in the theory of codes, digital signal processing, and statistical sampling. When the order m = 2s, they are called Sylvester matrices.
A Sylvester matrix has an equivalent matrix whose rows form a set of m-point Walsh functions or, in a different arrangement, Paley functions. Various linear Hadamard codes can be derived from a normalized Sylvester matrix in which +1 has been replaced by 0, and –1 by 1.
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