Moore-Penrose inverse in an indefinite inner product space (original) (raw)

The representation and approximation for the weighted Moore–Penrose inverse

Applied Mathematics and Computation, 2001

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. 0096-3003/02/$ -see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 -3 0 0 3 ( 0 2 ) 0 0 0 7 1 -1 Applied Mathematics and Computation 136 (2003) 475-486 www.elsevier.com/locate/amc

On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces

Filomat

For two given Hilbert spaces H and K and a given bounded linear operator A ? L(H,K) having closed range, it is well known that the Moore-Penrose inverse of A is a reflexive g-inverse G ? L(K,H) of A which is both minimum norm and least squares. In this paper, weaker equivalent conditions for an operator G to be the Moore-Penrose inverse of A are investigated in terms of normal, EP, bi-normal, bi-EP, l-quasi-normal and r-quasi-normal and l-quasi-EP and r-quasi-EP operators.

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

Applied Mathematics and Computation, 2010

The recursive method for computing the generalized LM -inverse of a constant rectangular matrix augmented by a column vector is proposed in . 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 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.

A generalization of the Moore–Penrose inverse related to matrix subspaces of

Applied Mathematics and Computation, 2010

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 A † S , is defined for any linear subspace S of the matrix vector space C n×m. The S-Moore-Penrose inverse A † S is characterized using either the singular value decomposition or (for the nonsingular square case) the orthogonal complements with respect to the Frobenius inner product. These results are applied to the preconditioning of linear systems based on Frobenius norm minimization and to the linearly constrained linear least squares problem.

On the Moore--Penrose inverse of a closed linear relation

Publicationes Mathematicae Debrecen, 2014

For a closed multivalued linear operator T between complex Hilbert spaces the concept of Moore-Penrose inverse of T , denoted T † , is introduced and studied. We prove that if y ∈ D(T † ), then T † y is the least square solution of minimal norm of the relation equation y ∈ T x. We also approximate T † by a sequence of bounded finite rank operators. Such results generalize the existing results to the case of densely defined closed operators.

On the Moore–Penrose generalized inverse matrix

2004

In this paper we exhibit some different methods for computing the Moore-Penrose inverse of different type of matrices. We discuss the Moore-Penrose inverse of block matrices, full-rank factorization. We give some numerical computations relative to this theory.

Maximal orthogonality and pseudo-orthogonality with applications to generalized inverses

Linear Algebra and its Applications, 2000

Equip a finite-dimensional vector space V over a field F with a nondegenerate symmetric bilinear, alternating bilinear, or hermitian form. Fix some subspace U of V. The concept of a maximally orthogonal complementary subspace for U is presented and shown to be related to the concept of a pseudo-orthogonal complementary subspace from an earlier paper. When F = GF(q), the number of maximally orthogonal (pseudo-orthogonal) complements is computed. Also, both types of complements are characterized as orthogonal complements with respect to related forms. This is used to relate certain reflexive generalized inverses of linear transformations to Moore-Penrose inverses. Techniques are presented for deriving the related forms, which, together with standard techniques for computing Moore-Penrose inverses, can be used to compute the desired generalized inverses. Lastly, vectors of V that occur in such complementary subspaces of U are characterized.

A revisitation of formulae for the Moore–Penrose inverse of modified matrices

Formulae for the Moore-Penrose inverse M + of rank-one-modifications of a given m × n complex matrix A to the matrix M = A + bc * , where b and c * are nonzero m × 1 and 1 × n complex vectors, are revisited. An alternative to the list of such formulae, given by Meyer [SIAM J. Appl. Math. 24 (1973) 315] in forms of subtraction-addition type modifications of A + , is established with the emphasis laid on achieving versions which have universal validity and are in a strict correspondence to characteristics of the relationships between the ranks of M and A. Moreover, possibilities of expressing M + as multiplication type modifications of A + , with multipliers required to be projectors, are explored. In the particular case, where A is nonsingular and the modification of A to M reduces the rank by 1, such a possibility was pointed out by Trenkler [R.D.H. Heijmans, D.S.G. Pollock, A. Satorra (Eds.), Innovations in Multivariate Statistical Analysis. A Festschrift for Heinz Neudecker, Kluwer, London, 2000, p. 67]. Some applications of the results obtained to various branches of mathematics are also discussed.