Underdetermined direction-of-arrival (DOA) estimation for quasi-stationary signals impinging on a uniform circular array (UCA) with M sensors is addressed in this paper. We apply the Khatri–Rao (KR) approach to the UCA and obtain a new signal model which is capable of providing OðMÞ sensors. Meanwhile, the virtual steering matrix can be decomposed into a product of characteristic matrix dependi...

In [P90] we proposed to employ Vandermonde and Hankel multipliers to transform into each other the matrix structures of Toeplitz, Hankel, Vandermonde and Cauchy types as a means of extending any successful algorithm for the inversion of matrices having one of these structures to inverting the matrices with the structures of the three other types. Surprising power of this approach has been demon...

The results on Vandermonde-like matrices were introduced as a generalization of polynomial Vandermonde matrices, and the displacement structure of these matrices was used to derive an inversion formula. In this paper we first present a fast Gaussian elimination algorithm for the polynomial Vandermonde-like matrices. Later we use the said algorithm to derive fast inversion algorithms for quasise...

In this paper we consider a particularly important case of 3D monogenic polynomials that are isomorphic to the integer powers of one complex variable (called pseudo-complex powers or pseudo-complex polynomials, PCP). The construction of bases for spaces of monogenic polynomials in the framework of Clifford Analysis has been discussed by several authors and from different points of view. Here ou...

Let K be an infinite field. We give polynomial time constructions of families of r-dimensional subspaces of Kn with the following transversality property: any linear subspace of Kn of dimension n− r is transversal to at least one element of the family. We also give a new NP-completeness proof for the following problem: given two integers n and m with n m and a n ×m matrix A with entries in Z, d...

In this paper we introduce a new class of what we shall call polynomial Vandermonde-like matrices. This class generalizes the polynomial Vandermonde matrices studied earlier by various authors, who derived explicit inversion formulas and fast algorithms for inversion and for solving the associated linear systems. A displacement structure approach allows us to carry over all these results to the...

In the recent paper “On certain Vandermonde determinants whose variables separate” [Linear Algebra and its Applications 449 (2014) pp. 17–27], there was established a factorized formula for some bivariate Vandermonde determinants (associated to almost square grids) whose basis functions are formed by Hadamard products of some univariate polynomials. That formula was crucial for proving a conjec...

By means of comp lete symmetric polynomials this paper gives a new proof for the Vander-monde determinant formula. Another alternative proof for this formula is obtained via the collocatio n matrices. It also gives a generalized relationship between the Vandermo nde, the Pascal and the Stirling matrices. A new app roach to obtain the explicit inverse of the Vandermo nde matrix is investigated. ...

In a variety of contexts, physicists study complex, nonlinear models with many unknown or tunable parameters to explain experimental data. We explain why such systems so often are sloppy: the system behavior depends only on a few "stiff" combinations of the parameters and is unchanged as other "sloppy" parameter combinations vary by orders of magnitude. We observe that the eigenvalue spectra fo...

A simple numerical algorithm based on the generalized Inverse Vandermore matrix for evaluation of time response to a time-invariant system is proposed. The system is assumed to be governed by a high order linear differential equation with constant coefficients. The technique involves determination of the partial-fraction expansion of rational functions. Only synthetic division and longhand divi...