Inversion Of Matrices Research Articles

Some Results on Core Inverses of Block Matrices Over Skew Fields

In this paper, necessary and sufficient conditions are given for the existence of the core inverse of the block matrix \(\begin{pmatrix}A&A\\B&0 \end{pmatrix}\) over any skew field, where \(A\), \(B\) are both square and \(rk(B) \geq rk(A)\). The representation of this core inverse and some relative additive results are also given.

Initialization and Estimation of Weights and Bias using Bayesian Technique

Researches on artificial neural network models have shown that the method used to initialize and estimate weights and bias always determines the rate at which the network will converge and how efficient the model will perform. Although there are several methods that can be used to initialize and estimate network weights and bias, Bayesian approach is widely considered as a more efficient method for modelling artificial neural networks because it can easily compute the inverses of covariance matrices with high dimension that otherwise are computationally expensive. This study has developed a new filter, the First order Extended Ensemble Filter(FoEEF),that applies numerical solution in solving the inverses of high dimensional covariance matrices from I ^ to ′ s stochastic state-space dynamical models. The research applies the new FoEEF Filter to initializes and estimates the weights and bias of artificial neural network. Comparison on the performance of FoEEF filter in estimating weights and bias are done against the performance of EKF on function estimations by using two different functions, sin(x)+Q function and 3sin(x)3− 3sin(x)2 − sin(x) + 1 function, within 18 and 22 epochs. Emphasis of the performance is placed on the rate at which the models converge. This study gauges the performance by determining the minimum value of the mean square error produced from each epoch and average mean square error minimum value.The outcome from the study showed that FoEEF filter had the lowest value of mean square error and average mean square error which was achieved with the least number of epochs compared to EKF filter. This study concluded that the new FoEEF performed better than EKF and is a more suitable filter for that can be used for initialization and estimation of weights and bias in artificial neural network.

The [formula omitted]-inverses: A generalization of core inverses for matrices

This paper concerns a new generalized inverse for matrices of an arbitrary index. It is proved that every complex square matrix A possesses a {1,2,3}-inverse X such that XAm+1=Am for some integer m. We shall call such X a {1,2,3,1m}-inverse of A. A notable result is that A has a unique {1,2,3,1m}-inverse if and only if it has index 0 or 1, in which case ▪ is exactly its unique {1,2,3,1m}-inverse. For a matrix with an arbitrary index, the set of all its {1,2,3,1m}-inverses is completely determined. Some new characterizations of EP matrices, generalized EP matrices and m-EP matrices are established by using their {1,2,3,1m}-inverses.

Fine‐grain task‐parallel algorithms for matrix factorizations and inversion on many‐threaded CPUs

AbstractWe extend a two‐level task partitioning previously applied to the inversion of dense matrices via Gauss–Jordan elimination to the more challenging QR factorization as well as the initial orthogonal reduction to band form found in the singular value decomposition. Our new task‐parallel algorithms leverage the tasking mechanism currently available in OpenMP to exploit “nested” task parallelism, with a first outer level that operates on matrix panels and a second inner level that processes the matrix either by ‐panels or by tiles, in order to expose a large number of independent tasks. We present a detailed performance analysis, including execution traces, which shows that the two‐level refinement into fine grain tasks allows for an improved load balancing and delivers high performance on current general‐purpose many‐core processors (CPUs) from Intel and AMD.

Group inverses of matrices associated with certain graph classes

We obtain formulae for group inverses of matrices that are associated with a new class of digraphs obtained from stars. This new class contains both bipartite and non-bipartite graphs. Expressions for the group inverse of matrices corresponding to double star digraphs and the adjacency matrix of certain undirected multi-star graphs are also proven. A blockwise representation of the inverse or group inverse of the adjacency matrix of the Dutch windmill graph is presented.

Materials Separation via the Matrix Method Employing Energy-Discriminating X-ray Detection

The majority of lab-based X-ray sources are polychromatic and are not easily tunable, which can make the 3D quantitative analysis of multi-component samples challenging. The lack of effective materials separation when using conventional X-ray tube sources has motivated the development of a number of potential solutions including the application of dual-energy X-ray computed tomography (CT) as well as the use of X-ray filters. Here, we demonstrate the simultaneous decomposition of two low-density materials via inversion of the linear attenuation matrices using data from the energy-discriminating PiXirad detector. A key application for this method is soft-tissue differentiation which is widely used in biological and medical imaging. We assess the effectiveness of this approach using both simulation and experiment noting that none of the materials investigated here incorporate any contrast enhancing agents. By exploiting the energy discriminating properties of the detector, narrow energy bands are created resulting in multiple quasi-monochromatic images being formed using a broadband polychromatic source. Optimization of the key parameters for materials separation is first demonstrated in simulation followed by experimental validation using a phantom test sample in 2D and a small-animal model in 3D.

New characterizations and representations of the generalized Bott–Duffin inverse and its applications

Some new characterizations of the generalized Bott–Duffin inverse are proposed by using range, nullspace and matrix equations. We extend the famous fact that the rank equation characterizes the inverse of matrices to the generalized Bott–Duffin inverse, and give its several representations. Additionally, we show the relationships between the generalized Bott–Duffin inverse and two classes of nonsingular bordered matrices and apply them to the Cramer's rules of two classes of restricted linear equations.

Perturbation bounds for the core inverse of matrices

Perturbation bounds for the core inverse of matrices

A note on formulae for the generalized Drazin inverse of anti-triangular block operator matrices in Banach spaces

A note on formulae for the generalized Drazin inverse of anti-triangular block operator matrices in Banach spaces

Efficient Identification of Multiple Parameter Errors in Power Grids by Mixed-Effects Models and Generalized Least Squares

Accurate knowledge of static parameters forms the basis for nearly all applications in the energy management system. This paper proposes an efficient method for the simultaneous identification and correction of multiple network parameter errors based on a linear mixed-effects (LME) model and the generalized least squares (GLS) method. An LME model for parameter error identification is formulated using the residual equations of multiple-snapshot state estimation with equality constraints. The parameter errors are considered as the fixed effects and the measurement errors are considered as the random effects. Then, using the measurement error variances estimated by the LME model, the GLS method is used to estimate the parameter errors along with a hypothesis testing to infer whether each parameter error is zero. The semi-supervised adversarial autoencoder is used for bad data detection in the presence of erroneous parameters and limited labels such that only measurement snapshots that are free of any bad data are used. The proposed methodology is efficient in that the LME model is only used to estimate the variances of the measurement errors using a small number of measurement snapshots, therefore the huge computation burden needed to solve a large-scale LME model is avoided. In addition, the GLS only involves inversion of low-dimension matrices, which is very efficient even a large number of measurement snapshots are used. Thorough tests of the proposed methodology on a large number of scenarios are provided to show the effectiveness of the proposed methodology with promising results.

Entries of the inverses of large positive definite Toeplitz matrices

AbstractThis is an expository paper embarking on the asymptotic behavior of the entries of the inverses of positive definite symmetric Toeplitz matrices as the matrix dimension goes to infinity. We consider the behavior of the entries in neighborhoods of the four corners as well as the density of the distribution of the entries over all of the inverse matrix.

Моделирование динамики упругих звеньев исполнительных механизмов манипуляторов без обращения их матриц масс

In recent scientific literature, much attention is paid to the optimal modeling of elastic dynamical systems. The relevance of these studies is dictated by the ever-increasing demand in the control theory of high-precision robotic manipulators and automatic mechanisms, which consists in the need for continuous adjustment of the movement of their executive bodies in real time, with account for the flexibility of the constituent links of these systems. The generalized Newton --- Euler method formulated in this regard served as a reliable platform for a subsequent progressive modification of dynamic analysis for a wide class of elastodynamic systems. The purpose of the study is to improve the existing computational algorithms to accelerate the computational process of dynamic analysis of flexible manipulators. In this regard, relying on symbolic-iterative calculus, we formulated and solved the mixed dynamic problem of the manipulators without inverting their mass matrices. Furthermore, we modified the Newton --- Raphson method, intended for a numerical integration of the Newton --- Euler equations of motion. Finally, we dynamically calculated the spatial five-elastic link manipulator by comparing the speed of computational processes carried out at constant accuracy of the problem modeling, and assessed the efficiency of the method of dynamic analysis of the manipulators. In this study, we introduce an advanced method of dynamic modeling of flexible manipulators without the well-known procedure of their mass matrices inversion

Public Key Cryptosystem Based on Singular Matrix

The algorithms such as RSA, ElGamal and ECC work on integers. Commutative operations on integer multiplication leave these algorithms vulnerable to attack by eavesdroppers. For this reason, experts develop the concept of non-commutative algebra in the public key cryptosystem by adding non-commutative properties to groups, semirings, semiring division, matrices and matrix decomposition. However, the key generating process in some public key cryptosystems is quite complicated to carry out. Therefore, in previous research, Liu used nonsingular matrices to form a simpler public key cryptosystem. However, eavesdroppers use the inverse of nonsingular matrices to construct the private key. As a result, this public key cryptosystem is still vulnerable to attacks. Therefore, we use a singular matrix to modify and build the proposed public key cryptosystem in this study. This study indicates that the singular matrix can be used to modify the public key cryptosystem. The results also show that the key generating algorithm only uses ordinary matrix multiplication (without using matrix power operations), so it is not too complicated. Furthermore, the proposed public key cryptosystem works on a matrix over integers so that the possible brute force attack trials are endless. The proposed public key cryptosystem also cannot be attacked by matrix inversion because it uses a singular matrix.
 HIGHLIGHTS
 
 Public key cryptosystems that use commutative operations are vulnerable to eavesdropping attacks
 This study uses a singular matrix to modify and build a public key cryptosystem
 The proposed public key cryptosystem works on ordinary matrix multiplication operations and cannot be attacked by matrix inversion

Solution of an inverse heat conduction problem with third-type boundary conditions

An algorithm is presented for the solution of an inverse heat conduction problem: reconstruct the steady-state distribution of the convection heat transfer coefficient on one wall of a slab from the temperature distribution on a plane embedded in the slab, assuming that the thermal boundary conditions on the other walls are known. The algorithm is based on the finite volume discretization of the slab and on the construction and subsequent inversion of square matrices linking the distributions of wall temperature and wall heat flux to that of temperature at the internal plane. The algorithm is first applied to a simulated temperature distribution, for which it is assessed for accuracy and compared with a simpler method based on the assumption of one-dimensional heat conduction. It is then applied to real experimental temperature distributions obtained by digitally processing thermochromic liquid crystals images in spacer-filled channels simulating membrane distillation modules. Tolerance to noise is demonstrated in both cases provided an adequate amount of filtering is applied to the primitive, high-resolution temperature distribution.

Schur Complement-Based Infinity Norm Bounds for the Inverse of GDSDD Matrices

Based on the Schur complement, some upper bounds for the infinity norm of the inverse of generalized doubly strictly diagonally dominant matrices are obtained. In addition, it is shown that the new bound improves the previous bounds. Numerical examples are given to illustrate our results. By using the infinity norm bound, a lower bound for the smallest singular value is given.

Further characterizations of the CMP inverse of matrices

In this paper, we give simple characterizations of the CMP inverse to be an EP matrix and simplify Theorem 2.10 in [New characterizations of the CMP inverse of matrices. Linear Multilinear Algebra. 2020;68(4):790–804]. Also, we characterize the CMP inverse of a matrix based on its range space and null space. Several different representations of the CMP inverse are given. In addition, some new characterizations of k-commutative equalities for some outer generalized inverse are investigated in terms of the CMP inverse.

Dual group inverses of dual matrices and their applications in solving systems of linear dual equations

<abstract><p>In this paper, we study a kind of dual generalized inverses of dual matrices, which is called the dual group inverse. Some necessary and sufficient conditions for a dual matrix to have the dual group inverse are given. If one of these conditions is satisfied, then compact formulas and efficient methods for the computation of the dual group inverse are given. Moreover, the results of the dual group inverse are applied to solve systems of linear dual equations. The dual group-inverse solution of systems of linear dual equations is introduced. The dual analog of the real least-squares solution and minimal $ P $-norm least-squares solution are obtained. Some numerical examples are provided to illustrate the results obtained.</p></abstract>

On the Drazin inverse of anti-triangular block matrices

<abstract><p>Our aim is to present new expressions for the Drazin inverse of anti-triangular block matrices under some circumstances. Applying the established new formulae for anti-triangular block matrices, we derive explicit representations for the Drazin inverse of a $ 2\times2 $ complex block matrix under corresponding assumptions. We extend several well known results in the literature in this way.</p></abstract>

Kernel Regression Over Graphs Using Random Fourier Features

This paper proposes efficient batch-based and online strategies for kernel regression over graphs (KRG). The proposed algorithms do not require the input signal to be a graph signal, whereas the target signal is defined over the graph. We first use random Fourier features (RFF) to tackle the complexity issues associated with kernel methods employed in the conventional KRG. For batch-based approaches, we also propose an implementation that reduces complexity by avoiding the inversion of large matrices. Then, we derive two distinct online strategies using RFF, namely, the mini-batch gradient KRG (MGKRG) and the recursive least squares KRG (RLSKRG). The stochastic-gradient KRG (SGKRG) is introduced as a particular case of the MGKRG. The MGKRG and the SGKRG are low-complexity algorithms that employ stochastic gradient approximations in the regression-parameter update. The RLSKRG is a recursive implementation of the RFF-based batch KRG. A detailed stability analysis is provided for the proposed online algorithms, including convergence conditions in both mean and mean-squared senses. A discussion on complexity is also provided. Numerical simulations include a synthesized-data experiment and real-data experiments on temperature prediction, brain activity estimation, and image reconstruction. Results show that the RFF-based batch implementation offers competitive performance with a reduced computational burden when compared to the conventional KRG. The MGKRG offers a convenient trade-off between performance and complexity by varying the number of mini-batch samples. The RLSKRG has a faster convergence than the MGKRG and matches the performance of the batch implementation.

New formulae of the Drazin inverse of anti-triangular complex block matrices

Let E, F ? Cnxn. If EFiE = 0 for all i ? N, we give the explicit representation of the Drazin inverse of the block complex matrix (E I F 0). We thereby solve a wider kind of singular differential equations posed by Campbell [S.L. Campbell, The Drazin inverse and systems of second order linear differential equations, Linear & Multilinear Algebra, 14(1983), 195-198].