46 research outputs found
On joint numerical radius of operators and joint numerical index of a Banach space
Generalizing the notion of numerical range and numerical radius of an
operator on a Banach space, we introduce the notion of joint numerical range
and joint numerical radius of tuple of operators on a Banach space. We study
the convexity of the joint numerical range. We show that the joint numerical
radius defines a norm if and only if the numerical radius defines a norm on the
corresponding space. Then we prove that on a finite-dimensional Banach space,
the joint numerical radius can be retrieved from the extreme points.
Furthermore, we introduce a notion of joint numerical index of a Banach space.
We explore the same for direct sum of Banach spaces. Applying these results,
finally we compute the joint numerical index of some classical Banach spaces.Comment: 17 page
S-Lemma with Equality and Its Applications
Let and be two quadratic functions
having symmetric matrices and . The S-lemma with equality asks when the
unsolvability of the system implies the existence of a real
number such that . The
problem is much harder than the inequality version which asserts that, under
Slater condition, is unsolvable if and only if for some . In this paper, we
show that the S-lemma with equality does not hold only when the matrix has
exactly one negative eigenvalue and is a non-constant linear function
(). As an application, we can globally solve as well as the two-sided generalized trust region subproblem
without any condition. Moreover, the
convexity of the joint numerical range where is a (possibly non-convex) quadratic
function and are affine functions can be characterized
using the newly developed S-lemma with equality.Comment: 34 page
Joint numerical range and its generating hypersurface
AbstractIn this paper, we study the joint numerical range of m-tuples of Hermitian matrices via their generating hypersurfaces. An example is presented which shows the invalidity of an analogous Kippenhahn theorem for the joint numerical range of three Hermitian matrices
On a multi-dimesional generalization of the notion of orthostochastic and unistochastic matrices
We introduce the notions of -orthostochastic, -unistochastic, and
-qustochastic matrices. These are the particular cases of -bistochastic
matrices where is real or complex numbers or quaternions. The concept is
motivated by mathematical physics. When , we recover the orthostochastic,
unistochastic, and qustochastic matrices respectively. This work exposes the
basic properties of -bistochastic matrices
A Fast Eigen Solution for Homogeneous Quadratic Minimization with at most Three Constraints
We propose an eigenvalue based technique to solve the Homogeneous Quadratic
Constrained Quadratic Programming problem (HQCQP) with at most 3 constraints
which arise in many signal processing problems. Semi-Definite Relaxation (SDR)
is the only known approach and is computationally intensive. We study the
performance of the proposed fast eigen approach through simulations in the
context of MIMO relays and show that the solution converges to the solution
obtained using the SDR approach with significant reduction in complexity.Comment: 15 pages, The same content without appendices is accepted and is to
be published in IEEE Signal Processing Letter