Weighted Moore-Penrose Inverse Research Articles

The representation and approximation for the weighted Minkowski inverse in Minkowski space

This paper extends some results for the weighted Moore–Penrose inverse A M , N + in Hilbert space to the so-called weighted Minkowski inverse A M , N ⊕ of an arbitrary rectangular matrix A ∈ M m , n in Minkowski spaces μ . Four methods are also used for approximating the weighted Minkowski Inverse A M , N ⊕ . These methods are: Borel summable, Euler–Knopp summable, Newton–Raphson and Tikhonov’s methods.

The representation and computation of generalized inverse [formula omitted

This paper presents a novel representation for the generalized inverse A T , S ( 2 ) . Based on this, we give an algorithm to compute this generalized inverse. As an application, we use Gauss–Jordan elimination to compute the weighted Moore–Penrose inverse A M , N † and the Drazin inverse A d .

Symbolic computation of weighted Moore–Penrose inverse using partitioning method

We propose a method and algorithm for computing the weighted Moore–Penrose inverse of one-variable rational matrices. Continuing this idea, we develop an algorithm for computing the weighted Moore–Penrose inverse of one-variable polynomial matrix. These methods and algorithms are generalizations of the method for computing the weighted Moore–Penrose inverse for constant matrices, originated in Wang and Chen [G.R. Wang, Y.L. Chen, A recursive algorithm for computing the weighted Moore–Penrose inverse A MN † , J. Comput. Math. 4 (1986) 74–85], and the partitioning method for computing the Moore–Penrose inverse of rational and polynomial matrices introduced in Stanimirović and Tasić [P.S. Stanimirović, M.B. Tasić, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137–163]. Algorithms are implemented in the symbolic computational package MATHEMATICA.

Vector least-squares solutions for coupled singular matrix equations

The weighted least-squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition. In this paper, a family of algorithms are applied to solve these problems based on the Kronecker structures. Subsequently, we construct a computationally efficient solutions of coupled restricted singular matrix equations. Furthermore, the need to compute the weighted Drazin and weighted Moore–Penrose inverses; and the use of Tian's work and Lev-Ari's results are due to appearance in the solutions of these problems. The several special cases of these problems are also considered which includes the well-known coupled Sylvester matrix equations. Finally, we recover the iterative methods to the weighted case in order to obtain the minimum D-norm G-vector least-squares solutions for the coupled Sylvester matrix equations and the results lead to the least-squares solutions and invertible solutions, as a special case.

Condition numbers with their condition numbers for the weighted moore-penrose inverse and the weighted least squares solution

In this paper, the authors investigate the condition number with their condition numbers for weighted Moore-Penrose inverse and weighted least squares solution of\(\mathop {\min }\limits_x ||Ax - b||_M \), whereA is a rank-deficient complex matrix in ℂm × n andb a vector of lengthm in ℂm,x a vector of length n in ℂn. For the normwise condition number, the sensitivity of the relative condition number itself is studied, the componentwise perturbation is also investigated.

An improved Newton iteration for the weighted Moore–Penrose inverse

In this paper, we showed that Newton iteration may be used to compute the weighted Moore–Penrose inverse of an arbitrary matrix. We also take advantage of several new acceleration procedures to reduce the cost of Newton iteration, which extends the earlier work [R. Schreiber, Computing generalized inverses and eigenvalues of symmetric matrices using systolic arrays, in: R. Glowinski, J.L. Lious (Eds.), Computing Methods in Applied Science and Engineering, North-Holland, Amsterdam, 1984; T. Söderstörm, G.W. Stewant, On the numerical properties of an iterative method for computing the Moore–Penrose generalized inverse, SIAM J. Numer. Anal. 11 (1974) 61–74].

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.

Some identities for Moore–Penrose inverses of matrix products

A group of identities are established for the Moore–Penrose inverses and the weighted Moore–Penrose inverses of matrix products AB and ABC. Some consequences and applications are also presented.

Some topics on weighted Moore–Penrose inverse, weighted least squares and weighted regularized Tikhonov problems

It is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems. The classical normwise relative condition number formulas are given by Higham [Linear Algebra Appl. 214 (1995) 193]. Here, we consider the condition number formulas for the weighted Moore–Penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems Ax= b. Moreover, we consider the nearness to rank deficiency. Finally, we give some decomposition tools and their application in the regularized weighted Tikhonov problems.

Rank equalities related to the generalized inverses AT, S(2), BT1, S1(2) of two matrices A and B

For given matrices A and B, by applying the group inverse expressions of generalized inverses A T, S (2) and B T 1, S 1 (2), we present some rank equalities related to A T, S (2) and B T 1, S 1 (2). As applications, we obtain a variety of rank equalities related to the Moore–Penrose inverses, the Drazin inverses, the group inverses and the weighted Moore–Penrose inverses of A and B.

The reverse order law for the generalized inverse A(2)T, S

By using the ranks of matrices, the necessary and sufficient condition is given for the reverse order law of the generalized inverse A (2) T, S of matrix product A= A 1 A 2⋯ A n to hold. As the special cases, similar results for Moore–Penrose inverse A † , the weighted Moore–Penrose inverse A † M,N, Drazin inverse A d and group inverse A g are also derived.

The weighted generalized inverses of a partitioned matrix

A constructive proof for the recursive representation of the weighted Moore–Penrose inverse of a partitioned matrix A=(U V) is given. Using the same thread of reasoning, similar results for A (1,3 M) and A (1,4 N) are derived. And all of these results can be represented by an unified form.

Using rank formulas to characterize equalities for Moore–Penrose inverses of matrix products

A group of rank formulas related to the Moore–Penrose inverses (AB) † and B †A † are established. Various interesting consequences of these rank formulas are obtained. The extensions of these results to the weighted Moore–Penrose inverses (AB) M,N † and B † P,NA † M,P are considered.

Condition numbers and perturbation of the weighted Moore–Penrose inverse and weighted linear least squares problem

It is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of nonsingular linear systems. Here, we consider the condition number formulas for the weighted Moore–Penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems Ax= b. These explicit expressions extend the earlier work of several authors. As we know, the computed weighted least squares solution x of min x ∥ Ax− b∥ M , in the presence of the round-off error, satisfies the perturbed equation min y ∥( A+ E) y−( b+ f)∥ M . Finally, an upper bound of ∥ y− x∥ N is derived for the case where the weighted matrix and vector norms and the assumption rank (A+E)= rank (A) are used.

A splitting iterative method for α− β generalized inverse and singular linear system

This paper deals with the case when the α− β generalized inverse A α, β (−1) is a linear transformation, in which case we give a splitting iterative method for A α, β (−1). We also show that in the case of linear transformation, the α− β generalized inverse A α, β (−1) is a {1,2}-inverse of the matrix A with prescribed range and null space, based on which we propose a iterative method of calculating the unique α-approximate solution of minimal β-norm of the system Ax= b. The results extend some previous results about the Moore–Penrose inverse A + and the weighted Moore–Penrose inverse A M, N +.

On integral representation of the generalized inverse AT, S(2)

We present a general integral representation for the generalized inverse A T, S (2), which extends earlier result on the Moore–Penrose inverse, weighted Moore–Penrose inverse and Drazin inverse.

The representation and approximation for the weighted Moore–Penrose inverse in Hilbert space

We present a unified representation theorem of the weighted Moore–Penrose inverse in Hilbert space. Specific expressions and computational procedures for the weighted Moore–Penrose inverse in Hilbert space can be uniformly derived.

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.

The weighted Moore–Penrose inverse of modified matrices

The weighted Moore–Penrose inverse of modified matrix A− CB is derived under some range conditions,where CB is a full rank factorization. The generalized singular value decomposition of a matrix is used to derive systematically the weighted Moore–Penrose inverse for a modified matrix, where CB need not be a full rank factorization. The results extend the earlier works by various authors.

The representation and approximation for the weighted Moore–Penrose inverse

We present a unified representation theorem for the weighted Moore–Penrose inverse. Specific expressions and computational procedures for the weighted Moore–Penrose inverse can be uniformly derived.