In several dimension reduction techniques, the original variables are replaced by a smaller number of linear combinations. The coefficients of these linear combinations are typically the elements of the left singular vectors of a random matrix. We derive the asymptotic distribution of the left singular vectors of a random matrix that has a normal limit distribution. This result is then used to ...

We present a novel setup that allows the observation of the geometric phase that accompanies polarization changes in monochromatic light beams for which the initial and final states are different (so-called non-cyclic changes). This Pancharatnam-Berry phase can depend in a linear or in a nonlinear fashion on the orientation of the optical elements, and sometimes the dependence is singular. Expe...

In this paper we examine the influence of dimensionality on natural language route directions in dialogue. Specifically, we show that giving route instructions in a quasi-3d environment leads to experiential descriptive accounts, as manifested by a higher proportion of location descriptions, lack of chunking, use of 1st person singular personal pronouns, and more frequent use of temporal and sp...

Given a complex matrix H, we consider the decomposition H=QRP∗ where Q and P have orthonormal columns, and R is a real upper triangular matrix with diagonal elements equal to the geometric mean of the positive singular values of H. This decomposition, which we call the geometric mean decomposition, has application to signal processing and to the design of telecommunication networks. The unitary...

Numerical and experimental realizations of quantum control are closely connected to the properties of the mapping from the control to the unitary propagator [14]. For bilinear quantum control problems, no general results are available to fully determine when this mapping is singular or not. In this paper we give sufficient conditions, in terms of elements of the evolution semigroup, for a traje...

T. Chan has noted that, even when the singular value decomposition of a matrix A is known, it is still not obvious how to find a rank-revealing QR factorization (RRQR) of A if A has numerical rank deficiency. This paper offers a constructive proof of the existence of the RRQR factorization of any matrix A of size m x n with numerical rank r. The bounds derived in this paper that guarantee the e...

For certain negative rational numbers κ0, called singular values, and associated with the symmetric group SN on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter κ = κ0. It was shown by the author, de Jeu and Opdam (Trans. Amer. Math. Soc. 346 (1994), 237-256) that the singular values are exactly the values − n with 2 ≤ n ≤ N , m = 1, 2, . . . ...