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This paper mainly studies some properties of normal matrix and gives the relation between the general solution of related matrix equations and normal matrices.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.

Image encryption based on elliptic curve cryptosystem and reducing its complexity is still being actively researched. Generating matrix for encryption algorithm secret key together with Hilbert matrix will be involved in this study. For a first case we will need not to compute the inverse matrix for the decryption processing cause the matrix that be generated in encryption step was self invertible matrix. While for the second case, computing the inverse matrix will be required. Peak signal to noise ratio (PSNR), and unified average changing intensity (UACI) will be used to assess which case is more efficiency to encryption the grayscale image.

This paper mainly introduces some equivalent conditions for SEP matrix, specifically by constructing some specific matrix equations and discussing whether these matrix equations have solutions in given set to determine whether a group invertible matrix is a SEP matrix.

In this paper, many interesting properties of normal matrices are given by means of such concepts as the power equivalities, projections, the regularity of vectors, one sided A-equality, A-commutativity and so on.

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

It is well known that, compared to low-dimension chaotic systems, three-dimensional chaotic systems have a wider parameter range, more complicated behaviour, and better unpredictability. This fact motivated us to introduce a novel image encryption method that employs a three-dimensional chaotic system. We proposed a novel three-dimensional conservative system that can exhibit chaotic behaviour involving hyperbolic functions. The dynamical behaviours of the proposed system are discovered by calculating Lyapunov exponents and bifurcation diagrams. Thereafter, we designed an image encryption method based on the proposed system and a 4×4 self-invertible matrix. A modified Diffie–Hellman key exchange protocol was utilised to generate the self-invertible key matrix Km employed in the diffusion stage. Our approach has three main stages. In the first stage, the proposed three-dimensional system utilises the original image to create three sequences, two of which are chosen for confusion and diffusion processes. The next stage involves confusing the image’s pixels by changing the positions of pixels using these sequences. In the third stage, the confused image is split into sub-blocks of size 4×4, and each block is encrypted by multiplying it with Km. Simulation findings demonstrated that the proposed image scheme has a high level of security and is resistant to statistical analysis, noise, and other attacks.

This paper mainly introduces some properties of several generalized inverses of matrices, especially some equivalent characteristics of generalized inverses of matrices, specifically by constructing some specific matrix equations and discussing whether these matrix equations have solutions in a given set to determine whether a group invertible matrix is some generalized inverse of matrices.

The theory of products of random matrices and Lyapunov exponents have been widely studied and applied in the fields of biology, dynamical systems, economics, engineering and statistical physics. We consider the product of an i.i.d. sequence of 2 × 2 random non-invertible matrices with real entries. Given some mild moment assumptions we prove an explicit formula for the Lyapunov exponent and prove a central limit theorem with an explicit formula for the variance in terms of the entries of the matrices. We also give examples where exact values for the Lyapunov exponent and variance are computed. An important example where non-invertible matrices are essential is the random Hill’s equation, which has numerous physical applications, including the astrophysical orbit problem.