196 research outputs found
The reflexive solutions of the matrix equation AX B = C
AbstractIn this paper, we study the existence of a reflexive, with respect to the generalized reflection matrix P, solution of the matrix equation AX B = C. For the special case when B = I, we get the result of Peng and Hu [1]
Solutions of the system of operator equations via -order
In this paper, we establish some necessary and sufficient conditions for the
existence of solutions to the system of operator equations in the
setting of bounded linear operators on a Hilbert space, where the unknown
operator is called the inverse of along . After that, under some
mild conditions we prove that an operator is a solution of if
and only if , where the -order means . Moreover we present the general solution
of the equation above. Finally, we present some characterizations of via other operator equations.Comment: 13 pages, to appear in Electron. J. Linear Algebra (ELA
Solving constrained Procrustes problems: a conic optimization approach
Procrustes problems are matrix approximation problems searching for
a~transformation of the given dataset to fit another dataset. They find
applications in numerous areas, such as factor and multivariate analysis,
computer vision, multidimensional scaling or finance. The known methods for
solving Procrustes problems have been designed to handle specific sub-classes,
where the set of feasible solutions has a special structure (e.g. a Stiefel
manifold), and the objective function is defined using a specific matrix norm
(typically the Frobenius norm). We show that a wide class of Procrustes
problems can be formulated and solved as a (rank-constrained) semi-definite
program. This includes balanced and unbalanced (weighted) Procrustes problems,
possibly to a partially specified target, but also oblique, projection or
two-sided Procrustes problems. The proposed approach can handle additional
linear, quadratic, or semi-definite constraints and the objective function
defined using the Frobenius norm but also standard operator norms. The results
are demonstrated on a set of numerical experiments and also on real
applications
Different invertibility modifications in operator spaces and c*-algebras and its applications
In this thesis different modifications of invertibility in various settings and their
applications are investigated. In particular, the reverse order law is considered for
classes of {1,3} and {1,4}-generalized inverses in C*-algebras and particulary in the
vector space of linear bounded operators on separable Hilbert spaces. The Hartwig's
triple reverse order law for Moore-Penrose inverse is discussed in C*-algebra and ring
with involution settings. The reverse order laws on {1,3}, {1,4}, {1,3,4}, {1,2,3} and
{1,2,4}-inverses in a ring setting are investigated. This results contain improvements
of some known results in C*-algebra case because the assumptions of the regularity of
some elements are omitted. The generalized invertibility is applied to solving certain
types of equations in rings with unit and determining the general form of solutions.
Strictly, the algebraic conditions for the existence of a solution and the expression for
the general solution of the system of three linear equations in a ring with a unit are
discussed. Another research concerns when the linear combinations of two operators
belonging to the class of Fredholm operators. Some cases where the Fredholmness of
linear combination is independent of the choice of the scalars are described in detail
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
HERIMITIAN SOLUTIONS TO THE EQUATION AXA* + BYB* = C, FOR HILBERT SPACE OPERATORS
Let A, A_{1}, A_{2}, B, B_{1}, B_{2}, C_{1} and C_{2} be linear bounded operators on Hilbert spaces. In this paper, by using generalized inverses, we establish necessary and sufficient conditions for the existence of a common solution and give the form of the general common solution of the operator equations A_{1}XB_{1}=C_{1} and A_{2}XB_{2}=C_{2}, we apply this result to determine new necessary and sufficient conditions for the existence of Hermitian solutions and give the form of the general Hermitian solution to the operator equation AXB=C. As a consequence, we give necessary and sufficient condition for the existence of Hermitian solution to the operator equation AXA^{*}+BYB^{*}=C
Some comments on the life and work of Jerzy K. Baksalary (1944-2005)
Following some biographical comments on Jerzy K. Baksalary (1944–2005), this article
continues with personal comments by Oskar Maria Baksalary, Tadeusz Cali´nski,
R.William Farebrother, Jürgen Groß, Jan Hauke, Erkki Liski, Augustyn Markiewicz,
Friedrich Pukelsheim, Tarmo Pukkila, Simo Puntanen, Tomasz Szulc, Yongge Tian,
Götz Trenkler, Júlia Volaufová, Haruo Yanai, and Fuzhen Zhang, on the life and
work of Jerzy K. Baksalary, and with a detailed list of his publications. Our article
ends with a survey by Tadeusz Cali´nski on Jerzy Baksalary’s work in block designs
and a set of photographs of Jerzy Baksalary
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