A natural generalization of the classical Moore-Penrose inverse is presented. The so-called S-Moore-Penrose inverse of a m × n complex matrix A, denoted by AS, is defined for any linear subspace S of the matrix vector space Cn×m. The S-Moore-Penrose inverse AS is characterized using either the singular value decomposition or (for the nonsingular square case) the orthogonal complements with resp...

In this paper, we present characterizations for the level-2 condition number of the weighted Moore–Penrose inverse, i.e., condMN (A) ≤ cond [2] MN (A) ≤ condMN (A)+ 1, where condMN (A) is the condition number of the weighted Moore–Penrose inverse of a rectangular matrix and cond [2] MN (A) is the level-2 condition number of this problem. This paper extends the result by Cucker, Diao and Wei [F....

Introduction. For a real m×n matrix A, the Moore–Penrose inverse A+ is the unique n×m matrix that satisfies the following four properties: AAA = A , AAA = A , (A+A)T = AA , (AA+)T = AA (see [1], for example). If A is a square nonsingular matrix, then A+ = A−1. Thus, the Moore–Penrose inversion generalizes ordinary matrix inversion. The idea of matrix generalized inverse was first introduced in ...

If A and B are a pair of invertible matrices of the same size, then the product AB is nonsingular, too, and the inverse of the product AB satisfies the reverse-order law (AB)−1 = B−1A−1. This law can be used to find the properties of (AB)−1, as well as to simplify various matrix expressions that involve the inverse of a matrix product. However, this formula cannot trivially be extended to the M...

In this article a fast computational method is provided in order to calculate the Moore-Penrose inverse of full rank m× n matrices and of square matrices with at least one zero row or column. Sufficient conditions are also given for special type products of square matrices so that the reverse order law for the Moore-Penrose inverse is satisfied.

In this article a fast computational method is provided in order to calculate the Moore-Penrose inverse of full rank m× n matrices and of square matrices with at least one zero row or column. Sufficient conditions are also given for special type products of square matrices so that the reverse order law for the Moore-Penrose inverse is satisfied.

We present some equivalent conditions of the reverse order law for the Moore–Penrose inverse in rings with involution, extending some well-known results to more general settings. Then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted Moore-Penrose inverse in C∗-algebras.

In this paper, we study representations of the Moore-Penrose inverse of a 2 × 2 matrix M over a ∗-regular ring with two term star-cancellation. As applications, some necessary and sufficient conditions for the Moore-Penrose inverse of M to have different types are given.

We investigate necessary and sufficient conditions for aae,f = bb † e,f to hold in rings with involution. Here, ae,f denotes the weighted Moore-Penrose inverse of a, related to invertible and Hermitian elements e, f ∈ R. Thus, some recent results from [7] are extended to the weighted Moore-Penrose inverse.

Abstraet--Jacobians are used in robotics for motion planning and control. They are also used in algorithms that determine linkage parameter errors of robots and in algorithms that determine pair variable corrections for accurate motion. Most applications require that the inverse of the Jacobian be obtained. The causes of singularities in Jacobians and a procedure to detect their presence are gi...