## Doctoral Theses

### Generalised Inverses of Matrices Over Rings.

2-28-1992

2-28-1993

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Abstract (Summary of the Work)

For a complex or a real matrix A, a matrix G is called a generalized inverse ( or g-inverse) of A if(1) AGA = AThe theory of generalized inverses over the field of complex numbers is well- studied in the literature (see (2), (11), (33), and (62) for an extensive bibliography). Even for matrices over a general field the above equation carries over. In fact, even for matrices over a general ring, equation (1) makes sense. Hence one can talk of g-inverses of matrices over general rings. Some work on g-inverses of matrices over fields also can be found in the literature.Our purpose in this thesis is to study g-inverses of matrices over rings. Our purpose in this thesis is to study g-inverses of matrices over rings. Over a commutative ring (even over an integral domain) because of the nonexistence of inverses for nonzero elements, the usual results on g-inverses of matrices over real or complex fields may not be extendable as discussed belowOver the real or complex field, more generally over any field, every matrix has a g-inverse. But even on the ring of integers, not every matrix has a g-inverse.For example, the matrix has no generalized inverse over Z. As early as 1939, von Neumann showed that every matrix over a ring A has a g-inverse if and only if A is regular.Another result which is true over any field is that every matrix over a field has a rank factorization. However, this is not true for a general integral domain. Similar observations lead us to a plethora of problems on g-inverses of matrices over general rings, in particular, over integral domains.Batigne in (4) gave necessary and sufficient conditions for integer matrices to have integer- g-inverses. Bose-Mitra (9) presented the first study of generalized inverses of polynomial matrices. These characterizations depend on the Smith normal form of matrices.However a matrix over a general integral domain need not have Smith normal form. For example, let D be the subring of the ring RCX, Y) of polynomials in X and Y with coefficients from the field of reals, generated by 1, X2, XY, and y2. The matrix has no rank factorization over D. Thus A does not have Smith normal form.Thus, if an integral domain is such that matrices over this integral domain do not admit Smith normal form, the results of (4) and [9) are not applicable and new techniques are required for study of g-inverses of matrices over such rings.Bose and Mitra in (9) and Sontag in (59) have observed that an important area of application of generalized inverses of matrices over general integral domains in System Science is investigation of under-determined and over- determined linear algebraic and differential systems. For example: a type of underdetermined systems is v = Cx where y is m x 1, C is m x n, and x is n x i and the elements of matrices are scalar valued functions of t defined over an interval 1. The above equation may be considered as the output equation of a control problem.

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