Least-squares inner product shaping
In this well structured and nicely written paper the author describes and proves several linear algebra results from his recent PhD thesis. He develops methods to 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 ones. 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 the constructed vectors is a circulant matrix. In the fourth, more general situation the rows of the Gram matrix of the constructed vectors are permutations of each other. The author mentions potential applications to a variety of problems, including a detection problem in quantum mechanics, noise elimination in signal processing and digital communication problems.
scientific article; zbMATH DE number 1760979
scientific article