Positive Definite Solutions of the Nonlinear Matrix Equation X+AHXˉ−1A=IX+A^{\mathrm{H}}\bar{X}^{-1}A=I

This paper is concerned with the positive definite solutions to the matrix equation $X+A^{\mathrm{H}}\bar{X}^{-1}A=I$ where $X$ is the unknown and $A$ is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation $W+B^{\mathrm{T}}W^{-1}B=I$ which has been extensively studied in the literature, where $B$ is a real matrix and is uniquely determined by $A.$ It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix $A$. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

Matris denklemleri ile ilişkili bazı özel tipli matrisler için matris yakınlık problemi

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Anahtar Kelimeler: minimum kalan problemi, matris yakınlık problemi, en iyi yaklaşık çözüm, Moore-Penrose ters. İlk bölümde lineer matris denklem problemleri ile ilgili literatür bilgisine yer verilmiş ve çalışmanın içeriğini oluşturan problemler tanıtılmıştır. İkinci bölümde çalışmada kullanılan bazı tanımlar ve temel teoremlerden bahsedilmiştir. Üçüncü bölümün ilk kısmında $\left( {{A_1}X{B_1},{A_2}X{B_2}, \ldots ,{A_k}X{B_k}} \right) = \left( {{C_1},{C_2}, \ldots ,{C_k}} \right)$ matris denkleminin simetrik ve ters-simetrik matrisler için genel çözümlerinin kümesi ve en küçük kareler çözümlerinin kümesi, Moore-Penrose ters ve Kronecker çarpım kullanılarak incelenmiştir. Bu matris denkleminin en iyi yaklaşık simetrik çözümü ve en iyi yaklaşık ters-simetrik çözümü ortaya konulmuştur. İkinci kısmında AXB=C matris denkleminin (P,Q)-ortogonal simetrik ve (P,Q)-ortogonal ters-simetrik matrisler için genel çözümlerinin kümesi ve en küçük kareler çözümlerinin kümesi Moore-Penrose ters ve spektral ayrışım kullanılarak incelenmiştir. Daha sonra, en iyi yaklaşık (P,Q)-ortogonal simetrik çözümü ve (P,Q)-ortogonal ters-simetrik çözümü elde edilmiştir. Son olarak, her iki kısmın sonunda ele alınan problemlerin çözümünü elde etmek için kullanılan bir algoritma, iki örnek ve literatürden seçilmiş örnekler için karşılaştırmalı bir tablo verilmiştir. Dördüncü bölümde (AXB,CXD)=(E,F) kuaterniyon matris denkleminin merkezi-hermityen ve ters-merkezi-hermityen matrisler üzerinde minimum kalan problemi Moore-Penrose ters, Kronecker çarpım ve vec operatörü kullanılarak incelenmiştir. Daha sonra ise (AXB,CXD)=(E,F) kuaterniyon matris denkleminin en iyi yaklaşık merkezi-hermityen çözümü ve ters-merkezi-hermityen çözümü verilmiştir. Son olarak, bölüm sonunda ele alınan problemlerin çözümünü elde etmek için kullanılan bir algoritma ve iki sayısal örnek verilmiştir. Son bölüm ise sonuçların kısa bir tartışmasına ayrılmıştır.Yeni kaotik sistemin FPGA tabalı tasarım

A theory of linear estimation

Theory of linear estimation and applicability to problems of smoothing, filtering, extrapolation, and nonlinear estimatio

Reliability in Constrained Gauss-Markov Models: An Analytical and Differential Approach with Applications in Photogrammetry

This report was prepared by Jackson Cothren, a graduate research associate in the Department of Civil and Environmental Engineering and Geodetic Science at the Ohio State University, under the supervision of Professor Burkhard Schaffrin.This report was also submitted to the Graduate School of the Ohio State University as a dissertation in partial fulfillment of the requirements for the Ph.D. degree.Reliability analysis explains the contribution of each observation in an estimation model to the overall redundancy of the model, taking into account the geometry of the network as well as the precision of the observations themselves. It is principally used to design networks resistant to outliers in the observations by making the outliers more detectible using standard statistical tests.It has been studied extensively, and principally, in Gauss- Markov models. We show how the same analysis may be extended to various constrained Gauss-Markov models and present preliminary work for its use in unconstrained Gauss-Helmert models. In particular, we analyze the prominent reliability matrix of the constrained model to separate the contribution of the constraints to the redundancy of the observations from the observations themselves. In addition, we make extensive use of matrix differential calculus to find the Jacobian of the reliability matrix with respect to the parameters that define the network through both the original design and constraint matrices. The resulting Jacobian matrix reveals the sensitivity of reliability matrix elements highlighting weak areas in the network where changes in observations may result in unreliable observations. We apply the analytical framework to photogrammetric networks in which exterior orientation parameters are directly observed by GPS/INS systems. Tie-point observations provide some redundancy and even a few collinear tie-point and tie-point distance constraints improve the reliability of these direct observations by as much as 33%. Using the same theory we compare networks in which tie-points are observed on multiple images (n-fold points) and tie-points are observed in photo pairs only (two-fold points). Apparently, the use of two-fold tiepoints does not significantly degrade the reliability of the direct exterior observation observations. Coplanarity constraints added to the common two-fold points do not add significantly to the reliability of the direct exterior orientation observations. The differential calculus results may also be used to provide a new measure of redundancy number stability in networks. We show that a typical photogrammetric network with n-fold tie-points was less stable with respect to at least some tie-point movement than an equivalent network with n-fold tie-points decomposed into many two-fold tie-points

Theoretical and Numerical Approaches to Co-/Sparse Recovery in Discrete Tomography

We investigate theoretical and numerical results that guarantee the exact reconstruction of piecewise constant images from insufficient projections in Discrete Tomography. This is often the case in non-destructive quality inspection of industrial objects, made of few homogeneous materials, where fast scanning times do not allow for full sampling. As a consequence, this low number of projections presents us with an underdetermined linear system of equations. We restrict the solution space by requiring that solutions (a) must possess a sparse image gradient, and (b) have constrained pixel values. To that end, we develop an lower bound, using compressed sensing theory, on the number of measurements required to uniquely recover, by convex programming, an image in our constrained setting. We also develop a second bound, in the non-convex setting, whose novelty is to use the number of connected components when bounding the number of linear measurements for unique reconstruction. Having established theoretical lower bounds on the number of required measurements, we then examine several optimization models that enforce sparse gradients or restrict the image domain. We provide a novel convex relaxation that is provably tighter than existing models, assuming the target image to be gradient sparse and integer-valued. Given that the number of connected components in an image is critical for unique reconstruction, we provide an integer program model that restricts the maximum number of connected components in the reconstructed image. When solving the convex models, we view the image domain as a manifold and use tools from differential geometry and optimization on manifolds to develop a first-order multilevel optimization algorithm. The developed multilevel algorithm exhibits fast convergence and enables us to recover images of higher resolution