Mitra, Sujit Kumar
(1968)
*A new class of g-inverse of square matrices*
Sankhya - Series A, 30
(3).
pp. 323-330.
ISSN 0581-572X

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Official URL: http://sankhya.isical.ac.in/search/30a3/30a3032.ht...

## Abstract

Necessary and sufficient conditions are obtained for a matrix A to have a g-inverse with rows and columns belonging to special linear manifolds. For a square matrix A, a g-inverse, with columns belonging to the linear manifold generated by the columns of A, is denoted by A^{-}_{C}. Such a g-inverse exists if and only if R(A)=R(A^{2}). The following properties of A^{-}_{C} are established: (a) A^{-}_{C}= A(A^{2})^{-}. (b) For any positive integer m, (A^{-}_{C})^{m} provides a reflexive g-inverse of A^{m}. (c) If x is an eigenvector corresponding to a nonnull eigenvalue λ of A, x is also an eigenvector of A^{-}_{C} corresponding to its eigenvalue 1/λ. The converse of this result is also true. (d) A special choice of (A^{2})^{-}=(A^{3})-A leads to A^{-}_{C}=A(A^{3})-A which is unique irrespective of the choice of (A^{3})^{-} and is, in fact, the same as the Scroggs-Odell pseudoinverse (J.SIAM 1966) of A. When R(A)=R(A^{2}), this indeed is a much simpler way of calculating the Scroggs-Odell pseudoinverse compared to the method indicated by its authors. (e) A(A^{3})-A belongs to the subalgebra generated by A.

Item Type: | Article |
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Source: | Copyright of this article belongs to Indian Statistical Institute. |

ID Code: | 32032 |

Deposited On: | 30 Mar 2011 12:53 |

Last Modified: | 30 Mar 2011 12:53 |

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