Simultaneously normalizable matrices

This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of ordern. Based on the proposed algorithm, the eigenvalues that have the largest and smallest modulus, real parts, or absolute values of imaginary parts can be extracted, respectively, as well as the corresponding eigenvectors. Although the ordinary differential equation on which our proposed algorithm is built is onlyn-dimensional, it can succeed to extractn-dimensional complex eigenvectors that are indeed 2n-dimensional real vectors. Moreover, we show that extracting eigen-pairs of general real matrices can be reduced to those of real normal matrices by employing the norm-reducing skill. Numerical experiments verified the computational capability of the proposed algorithm.

Shotgun metagenomics sequencing is a powerful tool for the characterization of complex biological matrices, enabling analysis of prokaryotic and eukaryotic organisms and viruses in a single experiment, with the possibility of reconstructing de novo the whole metagenome or a set of genes of interest. One of the main factors limiting the use of shotgun metagenomics on wide scale projects is the high cost associated with the approach. However, we demonstrate that—for some applications—it is possible to use shallow shotgun metagenomics to characterize complex biological matrices while reducing costs. We measured the variation of several summary statistics simulating a decrease in sequencing depth by randomly subsampling a number of reads. The main statistics that were compared are alpha diversity estimates, species abundance, detection threshold, and ability of reconstructing the metagenome in terms of length and completeness. Our results show that a classification of prokaryotic, eukaryotic and viral communities can be accurately performed even using very low number of reads, both in mock communities and in real complex matrices. With samples of 100,000 reads, the alpha diversity estimates were in most cases comparable to those obtained with the full sample, and the estimation of the abundance of all the present species was in excellent agreement with those obtained with the full sample. On the contrary, any task involving the reconstruction of the metagenome performed poorly, even with the largest simulated subsample (1M reads). The length of the reconstructed assembly was smaller than the length obtained with the full dataset, and the proportion of conserved genes that were identified in the meta-genome was drastically reduced compared to the full sample. Shallow shotgun metagenomics can be a useful tool to describe the structure of complex matrices, but it is not adequate to reconstruct—even partially—the metagenome.

The aim of this study is to investigate some properties of hybrid number matrices. Firstly, we introduce hybrid numbers with some of their properties. Then we prove that any hybrid number has a 2 ? 2 complex matrix representation, and we investigate hybrid number matrices using the properties of complex matrices. Also we give answers to the following basic question ?If =I, is it true that BA =I for hybrid number matrices?? Then we define the complex adjoint matrix and the q?determinant of hybrid number matrices and give some important properties. Finally, we give an explicit formula for the inverse of a hybrid number matrix by using complex matrices.

Impact of aging on nanoparticle toxicity in real matrices is scarcely investigated due to a lack of suitable methodologies. Herein, the toxicity of pristine and aged silver nanoparticles (Ag NPs) to a bioluminescent Pseudomonas putida bioreporter was measured in spiked crude and final wastewater samples (CWs and FWs, respectively) collected from four wastewater treatment plants (WWTPs). Results showed lower toxicity of pristine Ag NPs in CWs than in FWs. The effect of the matrix on the eventual Ag NP toxicity was related to multiple physico-chemical parameters (biological oxygen demand (BOD), chemical oxygen demand (COD), total suspended solids (TSS) pH, ammonia, sulfide and chloride) based on a multivariate analysis. However, no collection site effect was concluded. Aged Ag NPs (up to eight weeks) were found less toxic than pristine Ag NPs in CWs; evident increased aggregation and decreased dissolution were associated with aging. However, Ag NPs exhibited consistent toxicity in FWs despite aging; comparable results were obtained in artificial wastewater (AW) simulating effluent. The study demonstrates the potency of performing nanoparticle acute toxicity testing in real and complex matrices such as wastewaters using relevant bacterial bioreporters.

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.

The exponential function exp(x) = L' OXn/n! can be defined in any domain in which the infinite series converges; thus, in particular, it makes sense in any Banach algebra. This note is concerned with characterizing the range of exp in Mn(R), the set of all n X n real matrices, which is the simplest nontrivial real Banach algebra. Let us agree to write log B = A whenever exp A = B. Then we wish to determine which real matrices possess a real logarithm. It is elementary that for the domain R the range of exp is R + = { y > 0) and for the domain the range is C {O}. For the domain Mn(C), consisting of all n X n complex matrices, it is not too difficult to show that the range is GLn(C), the set of all nonsingular complex matrices. That it can be no larger than GLn(C) follows from the Jacobi identity det(exp A) = exp(tr A), a classical result which is also a consequence of remarks below. Standard infinite series expansions for the logarithm can be used to construct log B for certain large open sets in GLn(C). For example, if IIB -II < 1, where denotes any operator norm, then a logarithm can be defined by the series

Solving the matrix inversion is an open problem which is often related to scientific computation. Moreover, matrix inverse also has wide applications in social networks. Individuals in social networks are described as nodes, and the similarity among nodes are significant for link prediction. Usually, the problem of calculating similarities among nodes is converted to the problem of matrix inversion. With the increasing of the orders of matrices, traditional sequential algorithms are unable to meet the needs for the short calculation time. Although cluster systems can solve the inversion of large-scale matrices efficiently, the equipment cost and power consumption are very high. This paper proposes a parallel algorithm PA-Gauss, which is based on the Gauss-Jordan method of selecting the main element. CUDA (Computer Unified Device Architecture) of GPU (Graphic Process Unit) is used to implement the proposed algorithm to solve inversions of the real and complex matrices. The experimental results show that the Gauss-Jordan algorithm can save more running time than traditional sequential algorithms and the speedup ratio of PA-Gauss for Real Matrices is 633~100435, and the speedup ratio of PA-Gauss for Complex Matrices is 224~36508. Therefore,the computing time of solving the matrix inversions is reduced significantly.

The aim of this work is to offer necessary and sufficient conditions for the coincidence of the Drazin inverse and the DMP inverses of finite square complex matrices. This characterization is deduced from statements valid for finite potent endomorphisms and, in particular, several properties of matrices and generalized inverses are given.

In this paper I deal with quaternions history which were discovered by William Rowan Hamilton in 1843.&#x0D; Also I take some definitions and the basic properties of quaternions.&#x0D; My basic aim is how to representing quaternions as matrices. There are at least two ways of the representing of matrices, the first way is used 2×2 complex matrices and the second one is used 4×4 real matrices.&#x0D; I can able to generalize the first way by using n×n complex matrices that n is even.

The development of screen-printed electrochemical sensors represents a rapidly expanding research field with great potential for applications in the rapid and sensitive determination of drugs in complex matrices. This work presents a review of the state-of-the-art examples of this technology, focusing on its application in real matrices such as water, pharmaceutical formulations, and biological fluids. We discuss the main materials used in developing conductive inks, highlighting their properties and influence on sensor performance. The characterization of materials and sensors is crucial to ensure the reproducibility and reliability of results. Additionally, we address the challenges associated with the application of these sensors in complex matrices, such as interferences from other components and the need for sample pretreatment. Finally, we present future perspectives for developing screen-printed electrochemical sensors, with an emphasis on new technologies and materials that can improve the sensitivity, selectivity, and stability of these devices.

The bipolar decomposition

It is a fundamental problem in quantum information whether a particular quantum state of a composite system is entangled. It has enormous potential in quantum error correction, quantum cryptography, and quantum teleportation applications. This problem can be transferred in the form of a mathematical conjecture in language of linear algebra. In this paper, the authors explain the important applications, convenience, efficiency of using linear algebra in math physics, and computer science. The authors give some examples of linear algebra used in various areas, including datum coordinate system finding a location, encryption and decryption algorithm, storage of images, classical mechanics, and quantum physics. The authors list the definition of the matrix, real matrix, complex matrix, diagonal matrix, identity matrix, scalar matrix, trace, rank, and determinants of matrices. The authors explored Laplace expansion, transpose, inverse, conjugate of a matrix and addition, multiplication between matrices and between a scalar and a matrix, Kronecker product and their properties of matrices, the definitions and solving method of eigenvalue and eigenvector, and the diagonalisation and its conditions of matrices. The authors introduce the applications of matrix transformations and operations in programs. The authors explain two encryption and decryption algorithms based on linear algebra and their strengths and weaknesses.

In this paper, a canonical transformation is proposed to solve the eigenvalue problem related to the dynamics of rotor-bearing systems. In this problem, all matrices are real, but they may not be symmetric, which leads to the appearance of complex eigenvalues and eigenvectors. The bi-iteration method is selected to solve the original eigenproblem whereas the QR algorithm is adopted to solve the reduced or projected problem. A new canonical transformation of the global eigenproblem which reduces the quadratic eigenproblem to a linear eigenproblem, maintaining numerical stability since all that is required is that the stiffness matrix is well-conditioned, which is always true when it comes to applications in dynamic problems. The proposed technique is good for obtaining dominant eigenvalues and corresponding eigenvectors of real nonsymmetric matrices and it possesses the following properties: (i) the matrix is not transformed, therefore sparsity is maintained, (ii) partial eigensolutions can be obtained and (iii) use may be made of good eigenvectors predictions.

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi--Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

In this section, we seek to tie up some loose ends. First, although we have stated all the definitions and proved all the results in §§8–9 as if they applied only to real (n × n) matrices A, the truth is that just about everything said in §§8–9 about real matrices has its analogue in the theory of complex matrices. Let us go through §§8–9 point by point and isolate the (few) instances where realness of A makes a difference.