Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

Computing generalized inverses of a rational matrix and applications

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers [11] and [14]. and a continuation of the papers [15, 16]. The symbolic implementation of these algorithms in the package mathEmatica is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.     

On the Leverrier-Faddeev algorithm for computing the Moore-Penrose inverse

We present an improvement in the implementation of the Leverrier-Faddeev algorithm for symbolic computation of the Moore-Penrose inverse of one-variable polynomial matrices, introduced in Linear Algebra Appl. 252, 35–60 (1997). Complexity analysis of the original algorithm and its improvement is presented. Algorithm and its improvement are implemented and compared in the symbolic computational package MATHEMATICA. We compare CPU time required for computation of some test matrices by means of the original algorithm and its improvement.     

A finite algorithm for the Drazin inverse of a polynomial matrix

Based on Greville's finite algorithm for Drazin inverse of a constant matrix we propose a finite numerical algorithm for the Drazin inverse of polynomial matrices. We also present a new proof for Decell's finite algorithm through Greville's finite algorithm.     

On the Generalized Drazin Inverse and Generalized Resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C ^*-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.     

分块态射的广义逆

ClineRE给出了分块矩阵的Moore-Penrose逆的表达式,PetrPeska引进了分块态射的记号且导出了分块态射的Moore-Penrose逆的表达式.本文中,我们推广了Cline型分块态射的记号并得到了Cline型分块态射的Moore-Penrose逆和Drazin逆以及群逆的表达式.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

On generalized inverses and Green’s relations

We study generalized inverses on semigroups by means of Green’s relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class.     

Hilbert空间中算子广义逆的积分表示

利用算子矩阵分块的技巧,得到了Hilbert空间中算子的Moore-Penrose逆和Drazin逆的积分表示.给出了较为简洁的证明,同时将有限维的结论推广到无限维的情形.     

Some results on the Drazin inverse of anti-triangular matrices

Abstract

The representations for the Drazin inverse of anti-triangular matrices are obtained under some conditions. Applying these representations, we give a necessary condition for a class of block matrices to have signed Drazin inverse.     

About the generalized LM-inverse and the weighted Moore-Penrose inverse

The recursive method for computing the generalized LM-inverse of a constant rectangular matrix augmented by a column vector is proposed in Udwadia and Phohomsiri (2007) [16] and [17]. The corresponding algorithm for the sequential determination of the generalized LM-inverse is established in the present paper. We prove that the introduced algorithm for computing the generalized LM-inverse and the algorithm for the computation of the weighted Moore-Penrose inverse developed by Wang and Chen (1986) in [23] are equivalent algorithms. Both of the algorithms are implemented in the present paper using the package MATHEMATICA. Several rational test matrices and randomly generated constant matrices are tested and the CPU time is compared and discussed.     

An Imbedding Method for Computing the Generalized Inverses

This paper deals with a system of ordinary differential equations with known conditions associated with a given matrix. By using analytical and computational methods, the generalized inverses of the given matrix can be determined. Among these are the weighted Moore-Penrose inverse, the Moore-Penrose inverse, the Drazin inverse and the group inverse. In particular, a new insight is provided into the finite algorithms for computing the generalized inverse and the inverse.     

EP matrices: computation of the Moore-Penrose inverse via factorizations

EP matrices are a wide class of matrices which, among other things, can be characterized through factorizations. In this paper we are using two factorization algorithms in order to compute and factorize the Moore-Penrose inverse of a singular EP matrix. For the implementation of the algorithms we make use of a Computer Algebra System such as Maple. The results given by the proposed algorithms are faster and more accurate than the built-in Maple function in both symbolic and numerical tests, using matrices of different dimensions.     

Pseudo core inverses in rings with involution

Let R be a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse in R. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary ?-ring case by Raki?, Din?i? and Djordjevi? in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index in R. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of ?-ring. Some equivalent characterizations for elements in R to be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited.     

广义Bott-Duffin逆及加权Drazin逆的若干性质

本文给出了L-零矩阵的广义Bott-Duffin逆及矩阵的加权Drazin逆的若干性质及表达形式．     

The Moore-Penrose inverse for sums of matrices under rank additivity conditions

Some results on the Moore-Penrose inverse for sums of matrices under rank additivity conditions are revisited and some new consequences are presented. Their extensions to the weighted Moore-Penrose inverse of sums of matrices under rank additivity conditions are also considered.     

Interpolation algorithm of Leverrier–Faddev type for polynomial matrices

We investigated an interpolation algorithm for computing outer inverses of a given polynomial matrix, based on the Leverrier–Faddeev method. This algorithm is a continuation of the finite algorithm for computing generalized inverses of a given polynomial matrix, introduced in [11]. Also, a method for estimating the degrees of polynomial matrices arising from the Leverrier–Faddeev algorithm is given as the improvement of the interpolation algorithm. Based on similar idea, we introduced methods for computing rank and index of polynomial matrix. All algorithms are implemented in the symbolic programming language MATHEMATICA , and tested on several different classes of test examples.     

求鳞状因子循环矩阵的逆阵及广义逆阵的快速付氏变换法

借助快速付立叶变换(FFT),本文给出一种求n阶鳞状因子循环矩阵的逆阵、自反g-逆、群逆、Moore-Penrose逆的快速算法,该算法的计算复杂性为O(nlog2n),最后给出的两个数值算例表明了该算法的有效性.     

A generalization of biorthogonal systems based on Drazin inverse

In this paper, the transformations of generating systems of Euclidean space are examined in connection with the Drazin inverse of their Gram matrices and the notion of biorthogonal systems is generalized. The results obtained in this paper can be considered as the extension of some recent results concerning Moore-Penrose biorthogonal systems and reflexiveg-inverse systems in Euclidean spaces.