The theory of generalized inverses over commutative rings / K.P.S. Bhaskara Rao.

Format:

Book

Author/Creator:

Bhaskara Rao, K. P. S.

Contributor:

Class of 1891 Department of Arts Fund.

Series:

Algebra, logic, and applications ; v. 17.

Algebra, logic and applications ; v. 17

Language:

English

Subjects (All):

Linear operators--Generalized inverses.

Commutative rings.

Matrix inversion.

Physical Description:

xi, 167 pages ; 25 cm.

Place of Publication:

London ; New York : Taylor & Francis, 2002.

Contents:

1 Elementary results on rings 1

2 Matrix algebra over rings 5

2.1 Elementary notions 5

2.2 Determinants 8

2.3 The [partial differential]/[partial differential]a[subscript ij] notation 9

2.4 Rank of a matrix 10

2.5 Compound matrices 11

3 Regular elements in a ring 15

3.1 The Moore-Penrose equations 15

3.2 Regular elements and regular matrices 16

3.3 A Theorem of Von Neumann 19

3.4 Inverses of matrices 20

3.5 M-P inverses 25

4 Regularity - principal ideal rings 29

4.1 Some results on principal ideal rings 29

4.2 Smith Normal Form Theorem 32

4.3 Regular matrices over Principal Ideal Rings 38

4.4 An algorithm for Euclidean domains 41

4.5 Reflexive g-inverses of matrices 45

4.6 Some special integral domains 46

5 Regularity - basics 61

5.1 Regularity of rank one matrices 61

5.2 A basic result on regularity 62

5.3 A result of Prasad and Robinson 66

6 Regularity - integral domains 73

6.1 Regularity of matrices 73

6.2 All reflexive g-inverses 74

6.3 M-P inverses over integral domains 77

6.4 Generalized inverses of the form PCQ 82

6.5 {1,2,3}- and {1,2,4}-inverses 85

6.6 Group inverses over integral domains 89

6.7 Drazin inverses over integral domains 93

7 Regularity - commutative rings 101

7.1 Commutative rings with zero divisors 101

7.2 Rank one matrices 102

7.3 Rao-regular matrices 106

7.4 Regular matrices over commutative rings 114

7.5 All generalized inverses 119

7.6 M-P inverses over commutative rings 122

7.7 Group inverses over commutative rings 123

7.8 Drazin inverses over commutative rings 124

8 Special topics 127

8.1 Generalized Cramer Rule 127

8.2 A rank condition for consistency 129

8.3 Minors of reflexive g-inverses 130

8.4 Bordering of regular matrices 135

8.5 Regularity over Banach algebras 141

8.6 Group inverses in a ring 144

8.7 M-P inverses in a ring 146

8.8 Group inverse of the companion matrix 148.

Notes:

Includes bibliographical references (pages 153-165) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1891 Department of Arts Fund.

ISBN:

0415272483

OCLC:

48467137