Least-squares inner product shaping Original Research Article

We develop methods that construct an optimal set of vectors with a specified inner product structure, from a given set of vectors in a complex Hilbert space. The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. Four special cases are considered. In the first, the constructed vectors are orthonormal. In the second, they are orthogonal. In the third, the Gram matrix of inner products of the constructed vectors is a circulant matrix. As we show, the vectors form a cyclic set. In the fourth, the Gram matrix has the property that the rows are all permutations of each other. The constructed vectors are shown to be geometrically uniform.