The envelope, EðAÞ, of a complex square matrix A is a region in the complex plane that contains the spectrum of A and is contained in the numerical range of A. The envelope is compact but not necessarily convex or connected. The connected components of EðAÞ have the potential of isolating the eigenvalues of A, leading us to study its geometry, boundary, and number of components. We also examine...

Partially due to the fact that the empirical data collected by devices with finite bandwidth often neither preserves the specified structure nor induces a certain desired rank, retrieving the nearest structured low rank approximation from a given data matrix becomes an imperative task in many applications. This paper investigates the case of approximating a given target matrix by a real-valued ...

We present a nonperturbative solution of the rotational diffusion equation for an asymmetric rotor reorienting in a uniaxial liquid crystal, such as a nematic, smectic A or membrane bilayer covering the full range of order parameters from isotropic to complete order for rod-like and plate-like molecules. We obtain explicit expressions for the rotational diffusion matrix elements for a potential...

Let ‎X be an ‎‎n-‎‎‎‎‎‎square complex matrix with the ‎Cartesian decomposition ‎‎X = A + i ‎B‎‎‎‎‎, ‎where ‎‎A ‎and ‎‎B ‎are ‎‎‎n ‎‎times n‎ ‎Hermitian ‎matrices. ‎It ‎is ‎known ‎that ‎‎$Vert X Vert_p^2 ‎leq 2(Vert A Vert_p^2 + Vert B Vert_p^2)‎‎‎$, ‎where ‎‎$‎p ‎‎geq 2‎$‎ ‎and ‎‎$‎‎Vert . Vert_p$ ‎is ‎the ‎Schatten ‎‎‎‎p-norm.‎ ‎‎ ‎‎In this paper‎, this inequality and some of its improvements ...

Let Mn be the semigroup of n× n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mn including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of A ∈ Mn. Multiplicative maps φ : S → Mn satisfying rk(φ(A)...

This paper presents novel adaptive reduced-rank filtering algorithms based on joint iterative optimization of adaptive filters. The novel scheme consists of a joint iterative optimization of a bank of fullrank adaptive filters that constitute the projection matrix and an adaptive reduced-rank filter that operates at the output of the bank of filters. We describe least squares (LS) expressions f...

In Part I we proposed a multilinear algebra framework to solve 0-dimensional systems of polynomial equations with simple roots. We extend this incorporate multiple roots: block term decomposition (BTD) the null space Macaulay matrix reveals dual (sub)space disjoint root in each term. The BTD is joint triangularization multiplication tables and three-way generalization Jordan canonical form case...

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versi...