On the relation between Moore's and Penrose's conditions
Moore (1920) defined the reciprocal of any matrix over the complex field by three conditions, but the beauty of the definition was not realized until Penrose (1955) defined the same inverse using four conditions. The reciprocal is now often called the Moore-Penrose inverse, and has been widely used...
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| Format: | Article |
| Language: | English |
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202007317 |
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| _version_ | 1832545669608374272 |
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| author | Gaoxiong Gan |
| author_facet | Gaoxiong Gan |
| author_sort | Gaoxiong Gan |
| collection | DOAJ |
| description | Moore (1920) defined the reciprocal of any matrix over
the complex field by three conditions, but the beauty of the
definition was not realized until Penrose (1955) defined the same
inverse using four conditions. The reciprocal is now often called
the Moore-Penrose inverse, and has been widely used in
various areas. This note comments on the definitions of
Moore-Penrose inverse, and gives a new characterization for two
types of weak Moore-Penrose inverses, which exposes an
important relation between Moore's and Penrose's
conditions. It also attempts to emphasize the merit of Moore's
definition, which has been overlooked mainly due to Moore's
unique notation. Two examples are given to demonstrate some
combined applications of Moore's and Penrose's conditions,
including a correction for an incorrect proof of Ben-Israel's
(1986) characterization for Moore's conditions. |
| format | Article |
| id | doaj-art-be4b7ee75bbb4eca8a38f51b40a690b1 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-be4b7ee75bbb4eca8a38f51b40a690b12025-02-03T07:25:12ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0130850550910.1155/S0161171202007317On the relation between Moore's and Penrose's conditionsGaoxiong Gan0Department of Mathematics and Statistics, University of Missouri-Rolla, 1870 Miner Circle, Rolla 65409-0020, MO, USAMoore (1920) defined the reciprocal of any matrix over the complex field by three conditions, but the beauty of the definition was not realized until Penrose (1955) defined the same inverse using four conditions. The reciprocal is now often called the Moore-Penrose inverse, and has been widely used in various areas. This note comments on the definitions of Moore-Penrose inverse, and gives a new characterization for two types of weak Moore-Penrose inverses, which exposes an important relation between Moore's and Penrose's conditions. It also attempts to emphasize the merit of Moore's definition, which has been overlooked mainly due to Moore's unique notation. Two examples are given to demonstrate some combined applications of Moore's and Penrose's conditions, including a correction for an incorrect proof of Ben-Israel's (1986) characterization for Moore's conditions.http://dx.doi.org/10.1155/S0161171202007317 |
| spellingShingle | Gaoxiong Gan On the relation between Moore's and Penrose's conditions International Journal of Mathematics and Mathematical Sciences |
| title | On the relation between Moore's and Penrose's conditions |
| title_full | On the relation between Moore's and Penrose's conditions |
| title_fullStr | On the relation between Moore's and Penrose's conditions |
| title_full_unstemmed | On the relation between Moore's and Penrose's conditions |
| title_short | On the relation between Moore's and Penrose's conditions |
| title_sort | on the relation between moore s and penrose s conditions |
| url | http://dx.doi.org/10.1155/S0161171202007317 |
| work_keys_str_mv | AT gaoxionggan ontherelationbetweenmooresandpenrosesconditions |