Projections under seminorms and generalized Moore Penrose inverses

Mitra, Sujit Kumar ; Rao, C. Radhakrishna (1974) Projections under seminorms and generalized Moore Penrose inverses Linear Algebra and its Applications, 9 . pp. 155-167. ISSN 0024-3795

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/002437...

Related URL: http://dx.doi.org/10.1016/0024-3795(74)90034-2

Abstract

The definition of a projector under a seminorm is given. Such a projector is not unique. Operators projecting into a given linear subspace under a seminorm form an affine linear subalgebra of the linear associative algebra of square matrices. The authors have introduced elsewhere the concept of a minimum seminorm semileast squares inverse of a complex matrix. It is shown here that the same concept could also be defined in terms of projectors under seminorms. This extends a similar definition for the Moore Penrose inverse given in terms of orthogonal projectors under the usual Euclidean norms. Various properties of a projector under a seminorm and also of a minimum seminorm semileast squares inverse are obtained including representations giving general solutions for both.

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