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In this paper, we propose the generalized extreme learning machine (GELM). GELM is an ELM that incorporates the analyzed hyperparameters of ELM, such as sizes and ranks of weight matrices, and a limiting approach for the Moore–Penrose generalized inverse (M–P GI) into the learning process. ELM overcomes shortcomings of traditional deep learning, such as time-consuming due to iterative executions, as it learns quickly by removing the adjustment time of hyperparameters. There are desirable numbers of hidden nodes in ELM for single hidden layer feedforward neural networks, minimizing prediction error. However, it is difficult to use the desired number because it is related to the number of data used and datasets tend to be large. We consider ELM for multiple hidden layer feedforward neural networks. We analyze matrices derived in the network and figure out the characteristics of weight matrices and biases considering accurate prediction and learning speed, based on mathematical theories and a limiting approach for the M–P GI. The final output matrix of GELM is formulated explicitly. Experiments are conducted to verify the analysis using network traffic data, including DDoS attacks. The performances of GLEM, such as accuracies and learning speed, are compared for the networks with single and multiple hidden layers. Numerical results show the advantages of GELM in the performance measures, and the use of multiple hidden layers in GELM does not significantly affect performance. The theory-based prediction performances obtained from GELM will be the criterion for the margin of deep learning performance.

In this article, we generalize previously reported results for linear, time-invariant, stabilizable multivariable systems described by a strictly proper transfer function matrix $P(s)$ with number of outputs greater than or equal to the number of inputs. By making use of a special kind of a left generalized inverse $P(s)_{\alpha }^{\oplus }$ of $P(s)$ , we define and examine the equivalent relation $\mathcal {R}$ relating $P(s)$ with the members of the equivalence class $[P(s)]_{R}$ of the closed loop-transfer function matrices $P_{C}(s)$ obtainable from $P(s)$ by the use of a proper compensator $C(s)$ in the feedback path. For $\mathcal {R}$ , we establish a set of complete invariants and a canonical form. These results give rise to a simple algorithmic procedure for the computation of proper internally stabilizing and denominator assigning compensators $C(s)$ for the class of plants with $p=m$ and having no zeros in the closed right half complex plane: $\mathbb {C}^{+}$ and in the case when $p>m$ plants characterized by right polynomial matrix fraction descriptions with a polynomial matrix numerator having at least one subset of $m$ rows that give rise to a nonsingular polynomial matrix with no zeros in $\mathbb {C}^{+}$ .

Let $ J = \left[ \begin{array}{cc} 0 & I_n \\ -I_n & 0 \\ \end{array} \right] \in \mathbb{R}^{2n\times 2n} $. A matrix $ A \in \mathbb{R}^{2n\times 2n} $ is said to be Hamiltonian if $ (AJ)^{\top} = AJ $. In this paper, we first consider the following generalized inverse eigenvalue problem (GIEP): Given a pair of matrices $ (\Lambda, X) $ in the form $ \Lambda = {\rm{diag}}\{\lambda_1, \cdots, \lambda_{p}\}\in \mathbb{C}^{p\times p} $ and $ X = [{\bf{x}}_{1}, \cdots, {\bf{x}}_{p}]\in \mathbb{C}^{2n\times p} $, where diagonal elements of $ \Lambda $ are all distinct with rank$ (X) = p $, and both $ \Lambda $ and $ X $ are closed under complex conjugation in the sense that $ \lambda_{2i} = \bar{\lambda}_{2i-1}\in \mathbb{C}, $ $ {\bf{x}}_{2i} = \bar{{\bf{x}}}_{2i-1}\in \mathbb{C}^{2n} $ for $ i = 1, \cdots, l, $ and $ \lambda_{j}\in \mathbb{R}, $ $ {\bf{x}}_{j}\in \mathbb{R}^{2n} $ for $ j = 2l+1, \cdots, p. $ Find Hamiltonian matrices $ A $ and $ B $ such that $ AX\Lambda = BX. $ Then, we consider the associated optimal approximation problem (OAP): Given $ \tilde{A}, \tilde{B}\in \mathbb{R}^{2n\times 2n} $. Find $ (\hat{A}, \hat{B}) \in \mathbb{S_{E}} $ such that $ \|\hat{A}-\tilde{A}\|^2+\|\hat{B}-\tilde{B}\|^2 = {\min_{(A, B)\in \mathbb{S_{E}}}}\left(\|A-\tilde{A}\|^2+\|B-\tilde{B}\|^2\right), $ where $ \mathbb{S_{E}} $ is the solution set of Problem GIEP. By using the QR-decomposition, we deduce the representation of the general solution of Problem GIEP. Also, we obtain the unique optimal approximation solution $ (\hat{A}, \hat{B}) $ of Problem OAP.

A generalized inverse of an m × n matrix A over a pseudoring means an n × m matrix G satisfying AGA = A. In this paper we give a characterization of matrices having generalized inverses. Also, we introduce and study a space decomposition of a matrix, and prove that a matrix is decomposable if and only if it has a generalized inverse. Finally, we establish necessary and sufficient conditions for a matrix to possess various types of g-inverses including Moore–Penrose inverse.

A simple method to solve the common solution to the pair of linear matrix equations $ AXB = D $ and $ GXH = C $ is introduced. Some necessary and sufficient conditions for the existence of a common solution, and two expressions for the general common solution of the equation pair are provided by the proposed method. Subsequently, the results are applied to determine the solution of the matrix equation $ AXB+GYH = D $ and the Hermitian solution of the matrix equation $ AXB = D. $

Let A , B be N × N matrices such that A ≥ B ≥ 0 . We show that the matrix equation A = X + Y 2 , B = X # Y , 0 ≤ X ≤ Y has a unique solution X = A − ( A − B ) # ( A + B ) , Y = A + ( A − B ) # ( A + B ) , where the symbol ‘#’ denotes the operator geometric mean. We then determine every solution { X , Y } of the same equation under a weaker condition 0 ≤ X , 0 ≤ Y instead of 0 ≤ X ≤ Y . Such X and Y both satisfy the quadratic equation X A † X − 2 X + B A † B = 0 , where A † denotes the Moore-Penrose generalized inverse of A. Conversely, if X satisfies this equation, then so does Y : = 2 A − X and we have X # Y = B . We also state similar result regarding the equation A = X + Y 2 , C = X ! Y for given A ≥ C ≥ 0 , where ‘!’ denotes the operator harmonic mean.

Von Neumann regularity within a ring of upper triangular matrices is characterized in terms of how such a matrix and an upper triangular generalized inverse can be constructed. In the case of a fin...

[EN] This paper introduces and investigates a new class of generalized inverses, called GDMP-inverses (and their duals), as a generalization of DMP-inverses. GDMP-inverses are defined from G-Drazin inverses and the Moore-Penrose inverse of a complex square matrix. In contrast to most other generalized inverses, GDMP-inverses are not only outer inverses but also inner inverses. Characterizations and representations of GDMP-inverses are obtained by means of the core-nilpotent and the Hartwig-Spindelböck decompositions., Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (grant Resol. Nro. 135/19). Partially supported byMinisterio de Economia, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT). Secretaria de Estado de Investigacion, Desarrollo e Innovacion.

This paper focuses on state estimation problem for discrete-time high-order neural networks with time-varying delays. First, the delay-dependent global exponential stability criterion of the error system is derived. Then, the state observer is designed by using the generalized inverse theory of matrices. Last, two numerical examples are given to illustrate the validity of the theoretical results. The method proposed in this paper has two advantages: (i) it is directly based on the definitions of global exponential stability and Moore–Penrose inverse of matrix, which avoids the construction of Lyapunov–Krasovskii functional; (ii) the obtained stability criteria contain only several simple matrix inequalities, which are easier to solve. More valuable, this paper fills in the gaps in designing state observers for discrete-time high-order neural network models.

This article considers some fundamental estimation problems on a general partitioned linear model with partial parameter restrictions. We shall derive analytical formulas for calculating the ordina...

One of the fundamental research problems in the theory of generalized inverses of matrices is to establish reverse order laws for generalized inverses of matrix products, which are natural extensions of reverse order laws for the standard inverses of products of nonsingular matrices of the same size. Under the assumption that A, B, and C are singular matrices of the appropriate sizes, two reverse order laws for generalized inverses of the matrix products AB and ABC can be written as $$(AB)^{(i,\ldots ,j)} = B^{(s_2,\ldots ,t_2)}A^{(s_1,\ldots ,t_1)}$$ and $$(ABC)^{(i,\ldots ,j)} = C^{(s_3,\ldots ,t_3)}B^{(s_2,\ldots ,t_2)}A^{(s_1,\ldots ,t_1)}$$ , respectively, or other mixed reverse order laws. These equalities do not necessarily hold for different choices of generalized inverses of the matrices. Thus it is a tremendous work to classify and derive necessary and sufficient conditions for the reverse order laws to hold because there are all 15 types of $$\{i,\ldots , j\}$$ -generalized inverse for a given matrix according to the combinatoric choice of the four Penrose equations. In this paper, we first establish four groups of of mixed reverse order laws for $$\{1\}$$ - and $$\{1,2\}$$ -generalized inverses of AB and ABC. We then give a classified investigation to a family of reverse order laws $$(ABC)^{(i,\ldots ,j)} = C^{-1}B^{(s,\ldots ,t)}A^{-1}$$ for the eight commonly-used types of generalized inverses using the definitions of generalized inverses, the block matrix methodology, and the matrix rank methodology. A variety of consequences and applications of these reverse order laws are presented, and a list of open problems on reverse order laws are mentioned.

This paper discusses a nonlinear boundary value problem of system with rectangular coefficients of the form with boundary conditions of the form A(t)x' + B(t)x = f(t,x) and which is is a real matrix with whose entries are continuous on the form B1x(to)=a and B2x(T)=b which is A(t) is a real m n matri with m > n matrix with m > n whose entries are continuous on J = [to,T] and f E C[J x Rn, Rn]. B1, B2 are nonsingular matrices such that and are constant vectors, especially about the proof of the uniqueness of its solution. To prove it, we use Moore-Penrose generalized inverse and method of variation of parameters to find its solution. Then we show the uniqueness of it by using fixed point theorem of contraction mapping. As the result, under a certain condition, the boundary value problem has a unique solution.

Let A, B and C be $$n\times n$$ positive semi-definite matrices. We characterize $$C\le A ! B$$ , namely the harmonic mean of A, B, and apply it to show that if $$0

We introduce an algorithm for the least squares solution of a rectangular linear system $Ax=b$, in which $A$ may be arbitrarily ill-conditioned. We assume that a complementary matrix $Z$ is known such that $A - AZ^*A$ is numerically low rank. Loosely speaking, $Z^*$ acts like a generalized inverse of $A$ up to a numerically low rank error. We give several examples of $(A,Z)$ combinations in function approximation, where we can achieve high-order approximations in a number of non-standard settings: the approximation of functions on domains with irregular shapes, weighted least squares problems with highly skewed weights, and the spectral approximation of functions with localized singularities. The algorithm is most efficient when $A$ and $Z^*$ have fast matrix-vector multiplication and when the numerical rank of $A - AZ^*A$ is small.

In this paper, we define an operation on the intuitionistic fuzzy matrices called the Godel implication operator as an extension to the definition of this operator in the case of ordinary fuzzy matrices due to Sanchez and Hashimoto. Using this operator, we prove several important results for intuitionistic fuzzy matrices. Particularly, some properties concerning pre-orders, sub-inverses, and regularity . We concentrate our discussion on the reflexive and transitive matrices. This studying enables us to give a largest sub-inverse and a largest generalized inverse for a reflexive and transitive intuitionistic fuzzy matrix. Also, we obtain an idempotent intuitionistic fuzzy matrix from any given one.

In this paper, we propose a new hyperpower iterative method for approximating the weighted Moore-Penrose inverse of a given matrix. The main objective of the current work is to minimize the computational complexity of the hyperpower iterative method using some transformations. The proposed method attains the fifth-order of convergence using four matrix multiplications per iteration step. The theoretical convergence analysis of the method is discussed in detail. A wide range of numerical problems is considered from scientific literature, which demonstrates the applicability and superiority of the proposed method.

The system of matrix Lurie equations is considered. Such a system is of practical importance in the study of the asymptotic stability of equilibrium states of a system of differential equations, finding the regions of attraction of equilibrium states, determining the conditions for the existence of limit cycles for systems of differential equations, investigating global stability, hidden synchronization of phase and frequency-frequency self-tuning systems. It is known that the conditions for the solvability of the matrix Lurie equations are determined by the "Yakubovich-Kalman frequency theorem". To study nonlinear oscillations of phase systems, it becomes necessary to find solutions of the matrix Lurie equations. In this paper we consider the case when the matrix Lyapunov inequality, which is part of the Lurie equation, has a matrix with real eigenvalues, some of which may be zero. For such a case, necessary and sufficient conditions for the solvability of the Lurie equations are obtained and the form of the solutions is determined, which makes it possible to carry out their spectral analysis. The explicit form of the solutions of the matrix equations made it possible to make their geometric interpretation depending on the spectrum, to show the relationship of the linear connection equation to the quadratic forms of solutions of the Lurie equations. The method of analyzing matrix equations is based on an approach based on the use of a direct product of matrices and the application of generalized inverse matrices to find solutions to systems of linear equations. The results of the work made it possible to investigate the system of three matrix equations arising in the study of phase-frequency frequency-phase self-tuning systems.

We propose an algorithm for solving the basis pursuit problem min u ∈ C n { ∥ u ∥ 1 : A u = f } . Our starting motivation is the algorithm for compressed sensing, proposed by Qiao, Li and Wu, which is based on linearized Bregman iteration with generalized inverse. Qiao, Li and Wu defined new algorithm for solving the basis pursuit problem in compressive sensing using a linearized Bregman iteration and the iterative formula of linear convergence for computing the matrix generalized inverse. In our proposed approach, we combine a partial application of the Newton’s second order iterative scheme for computing the generalized inverse with the Bregman iteration. Our scheme takes lesser computational time and gives more accurate results in most cases. The effectiveness of the proposed scheme is illustrated in two applications: signal recovery from noisy data and image deblurring.

One of the fundamental research problems in the theory of generalized inverses has certainly been establishments of various matrix equalities that involve generalized inverses. A simplest form of s...

Although solvability conditions for a system of two linear equations are well-known even in the case of rings and, for three linear equations, in the case of matrices, in the case of four linear eq...

In this paper, we construct the real representation matrix of canonical hyperbolic quaternion matrices and give some properties in detail. Then, by means of the real representation, we study linear equations, the inverse and the generalized inverse of the canonical hyperbolic quaternion matrix and get some interesting results.

The aim of this work is to extend to finite potent endomorphisms the notion of the Drazin-Moore-Penrose inverse of a finite matrix. Several properties of this generalized inverse are proved that are also valid for some infinite matrices. Moreover, Drazin-Reflexive inverses associated with arbitrary reflexive generalized inverses of a finite potent endomorphism are introduced.

In this paper, we present functional definitions of all types of generalized inverses related to the {1}-inverse, which is a continuation of the work of Campbell and Meyer (2009). According to these functional definitions, we further derive novel representations for all types of generalized inverses related to the {1}-inverse in terms of the bases for R(A*), N(A) and N(A*). Based on these representations, we present the corresponding algorithm for computing various generalized inverses related to the {1}-inverse of a matrix and analyze the computational complexity of our algorithm for a constant matrix. Finally, we implement our algorithm and several known algorithms for symbolic computation of the Moore-–Penrose inverse in the symbolic computational package MATHEMATICA and compare their running times. Numerical experiments show that our algorithm outperforms these known algorithms when applied to compute the Moore–Penrose inverse of one-variable rational matrices, but is not the best choice for two-variable rational matrices in practice.

This paper discusses Green’s matrix of nonlinear boundary value problem of first-order differential system with rectangular coeffisients, especially about its properties. In this case, the differential equation of the form with boundary conditions of the form and which is a real matrix with whose entries are continuous on and . , are nonsingular matrices such that and are constant vectors. To get the Green’s matrix and the assosiated generalized Green’s matrix, we change the boundary condition problem into an equivalent differential equation by using the properties of the Moore-Penrose generalized inverse, then its solution is found by using method of variation of parameters. The last we prove that the defined matrices satisfy the properties of green’s function. The result is the corresponding the Green’s matrix and the assosiated generalized Green’s matrix have the property of Green’s functions with the jump-discontinuity.

The forward order law for generalized inverse often appears in linear algebra problems of some applied fields, which have attracted considerable attention and some interesting results have been obtained. In this paper, using the extremal ranks of the generalized Schur complement, we obtain the necessary and sufficient conditions for the forward order laws $$\begin{aligned} A_1\{1,2,3\}A_2\{1,2,3\}\cdots A_n\{1,2,3\}\subseteq (A_1A_2\cdots A_n)\{1,2,3\} \end{aligned}$$ and $$\begin{aligned} A_1\{1,2,4\}A_2\{1,2,4\}\cdots A_n\{1,2,4\}\subseteq (A_1A_2\cdots A_n)\{1,2,4\}. \end{aligned}$$

In this paper, we investigate the existence and representations of the generalized inverse AT,∗(2), AT,∗(1,2), A∗,S(2), A∗,S(1,2), AT,S(2), AT,S(1,2) of a matrix A over an associative ring, and giv...

In this paper we present a new method for solving an m × n fuzzy linear system (FLS), A X ˜ = Y ˜ , where the coefficient matrix A is real, using the block representation of generalized inverses. A necessary and sufficient condition for a block matrix to be the Moore–Penrose inverse of the full rank matrix associated to a FLS is given. We obtain a necessary and sufficient condition for the existence of solutions of a FLS, with arbitrary real coefficient matrix. The exact algebraic form, with respect to the Moore–Penrose inverse, of any solution of FLS of this type is established. A general, efficient and universal method for obtaining the exact solutions is introduced. Some numerical examples are presented to illustrate the proposed method.

International audience; Updating a linear least squares solution can be critical for near real-time signalprocessing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A ∈ R n×m with rank r. In this paper, we explicitly derive a similar formula by maintaining a general rank factorization, which we call rank-Greville. Based on this formula, we implemented a recursive least squares algorithm exploiting the rank-deficiency of A, achieving the update of the minimum-norm least-squares solution in O(mr) operations and, therefore, solving the linear least-squares problem from scratch in O(nmr) operations. We empirically confirmed that this algorithm displays a better asymptotic time complexity than LAPACK solvers for rank-deficient matrices. The numerical stability of rank-Greville was found to be comparable to Cholesky-based solvers. Nonetheless, our implementation supports exact numerical representations of rationals, due to its remarkable algebraic simplicity.

In this article, Special Infinite Matrix (SIM) and Complementary Special Infinite Matrix (CSIM) are introduced and its basic properties are studied. Various types of Symmetric Properties for SIM and CSIM are discussed. The existence of Generalized Inverse and Special Inverses for SIM and CSIM are derived. The complexity of SIM and CSIM are analyzed and suitable examples are provided.

Finding the maximum testing accuracy in Machine Learning has been the goal since its conception. From this goal, neural networks have been the primary source of continual improvements in prediction performance. Traditionally, backpropagation has been the primary way of training neural networks and the Levenberg-Marquardt (LM) backpropagation has become the fastest method. Recently, the Extreme Learning Machine was introduced which randomizes weights and biases of hidden layers and uses the Moore-Penrose generalized inverse of a matrix to calculate the output weights and biases, providing competitive results at significantly faster training times. In this study, we continue our work on the Constrained State-Preserved Extreme Learning Machine (CSPELM) with a Forest optimization (CSPELMF) and $\varepsilon$ constraint Rangefinder (CSPELMR). Furthermore, we provide hyper-parameter settings for the CSPELM to optimize accuracy over training time. Our results show that our methods outperformed the LM backpropagation in a majority of the 13 tested datasets and that the CSPELMF and CSPELMR matched or outperformed the CSPELM in all classification datasets.

Traditional expansion techniques utilize a modal projection wherein modal response is estimated based on a generalized inverse of measurements at a sparse set of degrees of freedom. Those modal response estimates are then used to project out to a larger set of degrees of freedom, resulting in predicted responses at more points or even full- field. As with any generalized inverse problem, the results are sensitive to noise and conditioning of the inverted matrix. While much has been done to improve numerics of matrix inversion problems in the context of input estimation or source identification problems, little has been done to improve the numerics of inverse solutions in expansion problems. This work presents numerical correction or regularization techniques applied to expansion problems using both simple and complex example structures. The effects of degree of freedom selection and noise are explored. Improved expansion results are obtained using straightforward regularization techniques, meaning higher accuracy responses can be obtained at expansion degrees of freedom with no change in the sparse set of measurements.

Reverse order laws for generalized inverses of matrix products are a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product AB is (AB)(i,…,j)=B(s2,…,t2)A(s1,…,t1), where (⋅)(i,…,j) denotes a {i,…,j}-generalized inverse of matrix. Because {i,…,j}-generalized inverse of a singular matrix is not unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. The aim of this paper is to give an overview of plenty of results concerning reverse order laws for {i,…,j}-generalized inverses of the product AB, from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model y=ABβ+Aγ+ϵ and the presentation of two least-squares methodologies for estimating the fixed parameter vector β in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of AB. We then prepare various necessary matrix study tools, including a general theory on linear or nonlinear algebraic matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of B(s2,…,t2)A(s1,…,t1), as well as necessary and sufficient conditions for B(s2,…,t2)A(s1,…,t1) to be invariant with respect to the choice of the generalized inverses, etc. Subsequently, we present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of A, B, and AB by means of the definitions of generalized inverses, the block matrix methodology (BMM), the matrix equation methodology (MEM), and the matrix rank methodology (MRM)., Mathematics; Matrix product; Orthogonal projector; Idempotent matrix; Generalized inverse; Reverse order law; Matrix-valued function; Linear statistical model; Ordinary least-squares estimator; Block matrix methodology; Matrix equation methodology; Matrix rank methodology

The M-P (Moore-Penrose) pseudoinverse has as a key application the computation of least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given input matrix is sparse, its M-P pseudoinverse can be dense, potentially leading to high computational burden, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but only two of them need to be satisfied for the computation of least-squares solutions. Fampa and Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy the two key M-P properties, plus one more (the so-called reflexive property). That additional M-P property is equivalent to imposing a minimum-rank condition on the generalized inverse. (Vector) 1-norm minimization is used to induce sparsity and, importantly, to keep the magnitudes of entries under control for the generalized-inverses constructed. Here, we investigate the trade-off between low 1-norm and low rank for generalized inverses that can be used in the computation of least-squares solutions. We propose several algorithmic approaches that start from a $1$-norm minimizing generalized inverse that satisfies the two key M-P properties, and gradually decrease its rank, by iteratively imposing the reflexive property. The algorithms iterate until the generalized inverse has the least possible rank. During the iterations, we produce intermediate solutions, trading off low 1-norm (and typically high sparsity) against low rank.

In this paper, the inconsistent LR fuzzy matrix equation A X ˜ = B ˜ is proposed and discussed. Firstly, the LR fuzzy matrix equation is transformed into two crisp matrix equations in which one determines the mean value and the other determines the left and right extends of fuzzy approximate solution. Secondly, the approximate solution of the LR fuzzy matrix equation is obtained by solving two crisp matrix equations according to the generalized inverse of crisp matrix theory. Then, sufficient conditions for the existence of strong LR fuzzy approximate solution are given. Finally, some numerical examples are given to illustrate our proposed method.

The purpose of this paper is to study a new generalized inverse called weighted CMP inverse associated with an operator between two Hilbert spaces, using its Wg-Drazin inverse and its Moore–Penrose inverse. This generalized inverse extends the notion of the weighted CMP inverse for a rectangular matrix. We give some new properties of weighted CMP inverse establishing matrix expression for the weighted CMP inverse of an operator and matrix expression for the Moore–Penrose inverse of this new inverse. Applying these results, we introduce and characterize the CMP inverse for a Hilbert space operator.

We study sparse generalized inverses H of a rank- r real matrix A . We give a construction for reflexive generalized inverses having at most r 2 nonzeros. For r = 1 and for r = 2 with A nonnegative, we demonstrate how to minimize the (vector) 1-norm over reflexive generalized inverses. For general r , we efficiently find reflexive generalized inverses with 1-norm within approximately a factor of r 2 of the minimum 1-norm generalized inverse.

This paper focuses on the parameter pattern during the initialization of Extreme Learning Machines (ELMs). According to the algorithm, model performance is highly dependent on the matrix rank of its hidden layer. Previous research has already proved that the sigmoid activation function can transform input data to a full rank hidden matrix with probability 1, which secures the stability of ELM solution. In recent study, we notice that, under full-rank condition, the hidden matrix possibly has very small eigenvalue, which seriously affects the model generalization ability. Our study indicates such a negative impact is caused by the discontinuity of generalized inverse at the boundary of full and waning rank. Experiments show that each phase of ELM modeling possibly leads to this rank deficient phenomenon, which harms the test accuracy.

This paper furnishes a general solution about the linear equation system $\boldsymbol{Ax}=\boldsymbol{g}$. The analytic solutions to the problem of finding the vector $\boldsymbol{x}$, from among the general solution set of the system if it is consistent, and from among the least squares solution set of the system if it is inconsistent, such that the norm of $\boldsymbol{x}-\boldsymbol{x}_{\mathbf{0}}$ is minimum for a given vector $\boldsymbol{x}_{\mathbf{0}}$ are established. For inverse matrix of A, it is used generalized inverse (Moore-Penrose inverse) by using algorithm and Maple. Analytic results, we obtained are satisfied by using algorithm with numerical examples.

This paper proposes a generalized inverse and mode assurance criterion based stochastic subspace identification (GMSSI) method to improve the electromechanical mode estimation efficiency in China Southern Power Grid (CSG). Stochastic subspace identification (SSI) is first employed to estimate the state subspace model of power system from synchrophasor measurements. Then, a model order determination criterion is proposed to estimate the model order and the state matrix of power system is obtained. To accurately separate the electromechanical modes from trivial modes, the generalized inverse of SSI is used to estimate another state matrix. Further, the mode filtering and mode assurance criterion (MAC) are developed to distinguish the electromechanical modes. Simulation data and field-measurement data from CSG are used to evaluate the performance of the proposed GMSSI method. The comparison between existing measurement-based mode estimation methods and the proposed GMSSI demonstrates that the proposed GMSSI exhibits highly accurate, efficient and robust performance in estimating electromechanical modes from both ringdown and ambient data in bulk power grid.

This paper studies the problem of designing the unknown input observers (UIOs) for fractional order one-sided Lipchitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using the matrix generalized inverse approach, sufficient conditions for asymptotic stability of the observer error dynamic systems are presented, which guarantee the existence of the full-order and reduced-order UIOs. All the conditions are obtained in terms of linear matrix inequality (LMI). Furthermore, we show that the obtained results can be applied to a fractional order electrical circuit with the unknown input signal. Two examples are given to demonstrate the applicability of the proposed approach.

In matrix theory and its applications, people often meet with various matrix expressions or matrix equalities that involve inverses of nonsingular matrices or generalized inverses of singular matrices. One of the fundamental research problems about these matrix expressions is to determine their singularity and nonsingularity, or alternatively, to determine upper and lower bounds of the ranks of the matrix expressions. Matrix rank formulas were introduced in the development of generalized inverse theory in 1970s, and many fundamental closed-form formulas for calculating ranks of matrices and their generalized inverses were established. In this paper, we consider the problem of determining the maximum and minimum ranks of a triple matrix product $$P^{(i,\ldots ,j)}NQ^{(i,\ldots ,j)}$$ , where P, Q, and N are given matrices, $$P^{(i,\ldots ,j)}$$ and $$Q^{(i,\ldots ,j)}$$ are $$\{i,\ldots , j\}$$ -generalized inverses of P and Q, respectively. We first rewrite the product as certain linear or nonlinear matrix-valued function with one or more variable matrices. We then establish a family of closed-form formulas for calculating the maximum and minimum ranks of $$P^{(i,\ldots ,j)}NQ^{(i,\ldots ,j)}$$ with respect to $$\{i,\ldots , j\}$$ -inverses of P and Q, respectively. Some applications of these max–min rank formulas are also discussed.

For the class of matrices over a field, the notion of `rank of a matrix' as defined by `the dimension of subspace generated by columns of that matrix' is folklore and cannot be generalized to the class of matrices over an arbitrary commutative ring. The `determinantal rank' defined by the size of largest submatrix having nonzero determinant, which is same as the column rank of given matrix when the commutative ring under consideration is a field, was considered to be the best alternative for the `rank' in the class of matrices over a commutative ring. Even this determinantal rank and the McCoy rank are not so efficient in describing several characteristics of matrices like in the case of discussing solvability of linear system. In the present article, the `rank--function' associated with the matrix as defined in [{\it Solvability of linear equations and rank--function}, K. Manjunatha Prasad, \url{http://dx.doi.org/10.1080/00927879708825854}] is discussed and the same is used to provide a necessary and sufficient condition for the existence of an outer inverse with specific column space and row space. Also, a rank condition is presented for the existence of Drazin inverse, as a special case of an outer inverse, and an iterative procedure to verify the same in terms of sum of principal minors of the given square matrix over a commutative ring is discussed.

We extend the notation of the CMP inverse for a square matrix to a rectangular matrix. Precisely, we define and characterize a new generalized inverse called the weighted CMP inverse. Also, we investigate properties of the weighted CMP inverse using a representation by block matrices. Some new characterizations and properties of the CMP inverse are obtained.

Following the work of Kentaro Nomakuchi[10] and Manjunatha Prasad et.al., [7] which relate various generalized inverses of a given matrix with suitable bordering,we describe the explicit bordering required to obtain core-EP inverse, core-EP generalized inverse. The main result of the paper also leads to provide a characterization of Drazin index in terms of bordering.

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J \in{\mathbb{R}}^{n\times n}$\end{document}J∈Rn×n be a normal matrix such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$J^{2}=-I_{n}$\end{document}J2=−In, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{n}$\end{document}In is an n-by-n identity matrix. In (S. Gigola, L. Lebtahi, N. Thome in Appl. Math. Lett. 48:36–40, 2015) it was introduced that a matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A \in{\mathbb {C}}^{n\times n}$\end{document}A∈Cn×n is referred to as normal J-Hamiltonian if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${(AJ)}^{*}=AJ$\end{document}(AJ)∗=AJ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$AA^{*}=A^{*}A$\end{document}AA∗=A∗A. Furthermore, the necessary and sufficient conditions for the inverse eigenvalue problem of such matrices to be solvable were given. We present some alternative conditions to those given in the aforementioned paper for normal skew J-Hamiltonian matrices. By using Moore–Penrose generalized inverse and generalized singular value decomposition, the necessary and sufficient conditions of its solvability are obtained and a solvable general representation is presented.

We propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop. This yields a high efficiency index for that computational task. The algorithm has a relatively mild numerical instability, and we stabilize it at the price of adding two extra matrix multiplications per iteration loop. This implies an efficiency index that exceeds the known record for numerically stable iterations for this task, which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra. Our numerical tests cover a variety of examples in the category of generalized inverses, such as Drazin case, rectangular case, and preconditioning of linear systems. The test results are in good accordance with our formal study.

Conventional extreme learning machines solve a Moore-Penrose generalized inverse of hidden layer activated matrix and analytically determine the output weights to achieve generalized performance, by assuming the same loss from different types of misclassification. The assumption may not hold in cost-sensitive recognition tasks, such as face recognition based access control system, where misclassifying a stranger as a family member may result in more serious disaster than misclassifying a family member as a stranger. Though recent cost-sensitive learning can reduce the total loss with a given cost matrix that quantifies how severe one type of mistake against another, in many realistic cases the cost matrix is unknown to users. Motivated by these concerns, this paper proposes an evolutionary cost-sensitive extreme learning machine (ECSELM), with the following merits: 1) to our best knowledge, it is the first proposal of ELM in evolutionary cost-sensitive classification scenario; 2) it well addresses the open issue of how to define the cost matrix in cost-sensitive learning tasks; 3) an evolutionary backtracking search algorithm is induced for adaptive cost matrix optimization. Experiments in a variety of cost-sensitive tasks well demonstrate the effectiveness of the proposed approaches, with about 5%~10% improvements., This paper has been accepted for publication in IEEE Transactions on Neural Networks and Learning Systems

This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse, an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

In this paper, we discuss the finite iterative algorithm to solve a class of generalized coupled Sylvester-conjugate matrix equations. We prove that if the system is consistent, an exact generalized Hamiltonian solution can be obtained within finite iterative steps in the absence of round-off errors for any initial matrices; if the system is inconsistent, the least squares generalized Hamiltonian solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the minimum norm least squares generalized Hamiltonian solution of the system. Finally, numerical examples are presented to demonstrate the algorithm is efficient.

The semipositive cone of $$A\in \mathbb {R}^{m\times n}, K_A = \{x\ge 0\,:\, Ax\ge 0\}$$ , is considered mainly under the assumption that for some $$x\in K_A, Ax>0$$ , namely, that A is a semipositive matrix. The duality of $$K_A$$ is studied and it is shown that $$K_A$$ is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.