Weighted Moore-Penrose Inverse Research Articles

Numerical range for weighted Moore-Penrose inverse of tensor

This article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define the notions of weighted normal tensor for an even-order square tensor and weighted tensor norm. Finally, we apply these to study the theory of numerical range for the weighted Moore-Penrose inverse of an even-order square tensor and exploit its several properties. We also obtain a few new results in matrix setting.

Computation of generalized inverses of tensors via t‐product

AbstractGeneralized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In this paper, we study different generalized inverses of tensors over a commutative ring and a noncommutative ring. Several numerical examples are provided in support of the theoretical results. We also propose algorithms for computing the inner inverses, the Moore–Penrose inverse, and weighted Moore–Penrose inverse of tensors over a noncommutative ring. The prowess of some of the results is demonstrated by applying these ideas to solve an image deblurring problem.

Weighted Moore-Penrose Inverses and Weighted Core Inverses in Rings with Involution

Weighted Moore-Penrose Inverses and Weighted Core Inverses in Rings with Involution

On (b,c)-inverses and (c,b)-inverses

We extend characterizations of inverses along an element to (b, c)-inverses. It is proved that if an element a in a semigroup S is both (b, c)-invertible and (c, b)-invertible for some then are group invertible, and is the (b, c)-inverse of a, is the (c, b)-inverse of a. Moreover, we establish several criteria for the existence of (b, c)-inverses and (c, b)-inverses by means of units in a ring. As applications, some new representations of weighted Moore-Penrose inverses and core-EP inverses of complex matrices are given.

A Novel Approach to Data Encryption Based on Matrix Computations

In this paper, we provide a new approach to data encryption using generalized inverses. Encryption is based on the implementation of weighted Moore–Penrose inverse A†MN(nxm) over the nx8 constant matrix. The square Hermitian positive definite matrix N8x8 p is the key. The proposed solution represents a very strong key since the number of different variants of positive definite matrices of order 8 is huge. We have provided NIST (National Institute of Standards and Technology) quality assurance tests for a random generated Hermitian matrix (a total of 10 different tests and additional analysis with approximate entropy and random digression). In the additional testing of the quality of the random matrix generated, we can conclude that the results of our analysis satisfy the defined strict requirements. This proposed MP encryption method can be applied effectively in the encryption and decryption of images in multi-party communications. In the experimental part of this paper, we give a comparison of encryption methods between machine learning methods. Machine learning algorithms could be compared by achieved results of classification concentrating on classes. In a comparative analysis, we give results of classifying of advanced encryption standard (AES) algorithm and proposed encryption method based on Moore–Penrose inverse.

Further Results on Weighted Core-EP Inverse of Matrices

In this paper, we introduce the notation of $E$-weighted core-EP and $F$-weighted dual core-EP inverse of matrices. We then obtain a few explicit expressions for the weighted core-EP inverse of matrices through other generalized inverses. Further, we discuss the existence of generalized weighted Moore-Penrose inverse and additive properties of the weighted core-EP inverse of matrices. In addition to these, we propose the star weighted core-EP and weighted core-EP star class of matrices for solving the system of matrix equations. We further elaborate on this theory by producing a few representation and characterization of star weighted core-EP and weighted core-EP star classes of matrices.

Weighted Moore–Penrose inverses of arbitrary-order tensors

Within the field of multilinear algebra, inverses and generalized inverses of tensors based on the Einstein product have been investigated over the past few years. The notion of the weighted Moore–Penrose inverses of even-order tensors in the framework of the Einstein product was introduced recently (Ji and Wei in Front Math China 12(6):1319–1337, 2017). In this article, we introduce the weighted Moore–Penrose inverse of an arbitrary-order tensor. We also investigate the singular value decomposition and full-rank decomposition of arbitrary-order tensors using reshape operation. Derived representations are used for two purposes: (1) to obtain a few new characterizations and representations of weighted Moore–Penrose inverse of arbitrary-order tensors; (2) to explore various necessary and sufficient conditions for the reverse-order law for the inverse to hold. In addition to these, we discuss applications of singular value decomposition and the Moore–Penrose inverse of an arbitrary-order tensor to a few 3D color image processing.

Weighted Moore–Penrose Inverses Associated with Weighted Projections on Indefinite Inner Product Spaces

Let H be a Hilbert $$C^*$$ -module, and let $$H_M$$ be the indefinite inner space induced by a self-adjointable and invertible operator M on H. Given weighted projections P and Q on $$H_M$$ , let $$S_{\lambda ,k}=(PQ)^k-\lambda (QP)^k$$ for a pair $$(k, \lambda )$$ , where k is a natural number and $$\lambda $$ is a complex number. It is proved that $$PQ-QP$$ is weighted Moore–Penrose invertible if and only if $$S_{\lambda ,k}$$ is weighted Moore–Penrose invertible for every pair $$(k, \lambda )$$ .

Weighted Moore–Penrose inverses of products and differences of weighted projections on indefinite inner-product spaces

An adjointable operator acting on a Hilbert $$C^*$$ -module is called a weight if it is self-adjoint and invertible. An indefinite inner-product space as well as weighted projections can be induced by a weight. Some new formulas are provided for weighted Moore–Penrose inverses associated to products and differences of weighted projections. As a result, some characterizations of Moore–Penrose inverses associated to projections are generalized to the weighted case.

Idempotents generated by weighted generalized inverses in rings with involution

Let R be an associate ring with unity 1 and involution. In this paper, we investigate several properties and characterizations of idempotents generated by weighted Moore-penrose inverses and weighted pseudo core inverses in R. Moreover, several new characterizations about weighted EP elements and existence criteria of weighted pseudo core inverses are also given.

An efficient hyperpower iterative method for computing weighted MoorePenrose inverse

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.

Reverse-order law for weighted Moore–Penrose inverse of tensors

The weighted Moore–Penrose inverse of a tensor via the Einstein product is introduced in the literature, very recently. The objective of this paper is to study the reverse-order law for the weighted Moore–Penrose inverse. In this context, a few properties of the weighted Moore–Penrose inverse of an arbitrary order tensor via the Einstein product are discussed. Some new sufficient conditions for the reverse-order law of even-order square tensors are obtained. Several characterizations of the reverse-order law for the weighted Moore–Penrose inverse of tensors via the same product are also presented.

Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration Scheme

The goal of this research is to extend and investigate an improved approach for calculating the weighted Moore–Penrose (WMP) inverses of singular or rectangular matrices. The scheme is constructed based on a hyperpower method of order ten. It is shown that the improved scheme converges with this rate using only six matrix products per cycle. Several tests are conducted to reveal the applicability and efficiency of the discussed method, in contrast with its well-known competitors.

Complex ZNN for computing time-varying weighted pseudo-inverses

We classify, extend and unify various generalizations of weighted Moore-Penrose inverses in indefinite inner product spaces. New kinds of generalized inverses are introduced for this purpose. These generalized inverses are included in the more general class called as the weighted indefinite pseudoinverses (WIPI), which represents an extension of the Minkowski inverse (MI), the weighted Minkowski inverse (WMI), and the generalized weighted Moore- Penrose (GWM-P) inverse. The WIPI generalized inverses are introduced on the basis of two Hermitian invertible matrices and two Hermitian involuntary matrices and represented as particular outer inverses with prescribed ranges and null spaces, in terms of appropriate full-rank and limiting representations. Application of introduced generalized inverses in solving some indefinite least squares problems is considered. New Zeroing Neural Network (ZNN) models for computing the WIPI are developed using derived full-rank and limiting representations. The convergence behavior of the proposed ZNN models is investigated. Numerical simulation results are presented.

Outer and (b,c) inverses of tensors

ABSTRACT Some basic properties of the range and null space of multidimensional arrays (tensors) with respect to Einstein tensor product are investigated. Also, an useful and effective definition of the tensor rank, termed as reshaping rank, is introduced. Further, computation of tensor outer with prescribed range and/or kernel of higher order tensors is considered. Starting from an algebraic approach, some new relationships between the problems of solving tensor equations and computation of tensor outer inverses with prescribed range and/or null pace are established. Conditions for the existence, representation and computation of the Moore-Penrose inverse, the weighted Moore-Penrose inverse, the Drazin inverse and the usual inverse of tensors are derived as corollaries. In this way, we derive extensions of known representations of various classes of matrix generalized inverses to multidimensional arrays. Effective algorithms for computing inner inverses of tensors as well as initiated algorithms for computing outer inverses of tensors are presented. Implementation of proposed algorithms in Matlab are developed and illustrative numerical examples are given. In addition, results related with the -inverses on semigroups are examined in details in a specific semigroup of tensors with a binary associative operation defined as the Einstein tensor product.

An interval extension of SMS method for computing weighted Moore–Penrose inverse

An interval extension of successive matrix squaring (SMS) method for computing the weighted Moore---Penrose inverse $$A^{\dagger }_{MN}$$AMN? along with its rigorous error bounds is proposed for given full rank $$m \times n$$m?n complex matrices A, where M and N be two Hermitian positive definite matrices of orders m and n, respectively. Starting with a suitably chosen complex interval matrix containing $$A^{\dagger }_{MN}$$AMN?, this method generates a sequence of complex interval matrices each enclosing $$A^{\dagger }_{MN}$$AMN? and converging to it. A new method is developed for constructing initial complex interval matrix containing $$A^{\dagger }_{MN}$$AMN?. Convergence theorems are established. The R-order convergence is shown to be equal to at least l, where $$l \ge 2$$l?2. A number of numerical examples are worked out to demonstrate its efficiency and effectiveness. Graphs are plotted to show variations of the number of iterations and computational times compared to matrix dimensions. It is observed that ISMS is more stable compared to SMS.

Computation of weighted Moore–Penrose inverse through Gauss–Jordan elimination on bordered matrices

In this paper, two new algorithms for computing the Weighted Moore–Penrose inverse AM,N† of a general matrix A for weights M and N which are based on elementary row and column operations on two appropriate block partitioned matrices are introduced and investigated. The computational complexity of the introduced two algorithms is analyzed in detail. These two algorithms proposed in this paper are always faster than those in Sheng and Chen (2013) and Ji (2014), respectively, by comparing their computational complexities. In the end, an example is presented to demonstrate the two new algorithms.

Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation

Weighted singular value decomposition of a quaternion matrix and with its help determinantal representations of the quaternion weighted Moore–Penrose inverse have been derived recently by the author. In this paper, using these determinantal representations, explicit determinantal representation formulas for the solution of the restricted quaternionic matrix equations, $$\mathbf{A}{} \mathbf{X}{} \mathbf{B}=\mathbf{D}$$ , and consequently, $$\mathbf{A}{} \mathbf{X}=\mathbf{D}$$ and $$\mathbf{X}{} \mathbf{B}=\mathbf{D}$$ are obtained within the framework of the theory of column–row determinants. We consider all possible cases depending on weighted matrices.

Multiplicative perturbations of matrices and the generalized triple reverse order law for the Moore–Penrose inverse

For a matrix A, let A† denote its Moore–Penrose inverse. A matrix M is called a multiplicative perturbation of T∈Cm×n if M=ETF⁎ for some E∈Cm×m and F∈Cn×n. Based on the alternative expression for M as M=(ETT†)⋅T⋅(FT†T)⁎, the generalized triple reverse order law for the Moore–Penrose inverse is obtained asM†=((FT†T)⁎)†⋅(YY†TZZ†)LR−1†⋅(ETT†)†, where (YY†TZZ†)LR−1† is the weighted Moore–Penrose inverse for certain matrices Y,Z,L and R associated to the triple (T,E,F). Furthermore, it is proved that this weighted Moore–Penrose inverse in the resulting expression for M† can be really replaced with T† if(ETT†)†ETT†⋅T=T⋅(FT†T)†(FT†T). In the special case that rank(M)=rank(T) or M is a weak perturbation of T, a simplified version of M†, as well as MM† and M†M, is also derived.

Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore–Penrose inverse

Weighted singular value decomposition (WSVD) and a representation of the weighted Moore–Penrose inverse of a quaternion matrix by WSVD have been derived. Using this representation, limit and determinantal representations of the weighted Moore–Penrose inverse of a quaternion matrix have been obtained within the framework of the theory of noncommutative column-row determinants. By using the obtained analogs of the adjoint matrix, we get the Cramer rules for the weighted Moore–Penrose solutions of left and right systems of quaternion linear equations.