@MISC{Henderson74someinvariant, author = {Keith W. Henderson}, title = {SOME INVARIANT PROPERTIES OF THE REAL HADAMARD MATRIX*}, year = {1974} }

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Abstract

Applications of well-known matrix theory reveal some interesting and possibly useful invariant properties of the real Hadamard matrix and transform (including the Walsh matrix and transform). Subject to certain conditions that can be fulfilled for many orders of the matrix, the space it defines can be decomposed into two invariant subspaces defined by two real, singular, mutually orthogonal (although not self-orthogonal) matrices, which differ from the Hadamard matrix only on the principal diagonal. They are their own Hadamard transforms, within a scalar multiplier, so their columns (or rows) are the eigenvectors of the Hadamard matrix. This relationship enables us to determine the eigenvalues of the Hadamard matrix, and to construct its Jordan normal form. The transformation matrix for converting the one to the other is equal to the sum of the Hadamard matrix and the Jordan form matrix. Unfortunately, therefore, it appears to be more complicated structurally than the Hadamard matrix itself, and does not lead to a simple method of generating the Hadamard matrix directly from its easily constructed Jordan form. Work supported by the Department of Energy. (Submitted for publication.)-2-