196 research outputs found

    The reflexive solutions of the matrix equation AX B = C

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    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 BXA=B=AXBBXA=B=AXB via *-order

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    In this paper, we establish some necessary and sufficient conditions for the existence of solutions to the system of operator equations BXA=B=AXB BXA=B=AXB in the setting of bounded linear operators on a Hilbert space, where the unknown operator XX is called the inverse of AA along BB. After that, under some mild conditions we prove that an operator XX is a solution of BXA=B=AXB BXA=B=AXB if and only if BAXAB \stackrel{*}{ \leq} AXA, where the *-order CDC\stackrel{*}{ \leq} D means CC=DC,CC=CDCC^*=DC^*, C^*C=C^*D. Moreover we present the general solution of the equation above. Finally, we present some characterizations of CDC \stackrel{*}{ \leq} D via other operator equations.Comment: 13 pages, to appear in Electron. J. Linear Algebra (ELA

    Solving constrained Procrustes problems: a conic optimization approach

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    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

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    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

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    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

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    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)

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    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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