## Doctoral Theses

### Generalized Inverses of Special Types of Matrices.

2-28-1974

2-28-1975

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Research and Training School (RTS)

#### Supervisor

Rao, C. Radhakrishna (RTS-Kolkata; ISI)

#### Abstract (Summary of the Work)

Inverses, in the regular sense of the term, do not exist for singular square matrices and rectangular matrices. however for such matrices there exist matrices which satisfy many important properties similar to those of inverses of nonsingular matrices and for many purposes, can be used in the same way as regular inverses. These matrices are named generalised inverses (g-inverses) to distinguish then from the inverses of nonsingular matricos. Only since 1955 this field of study af generalized inverse was: invostigated systenatically and was expiorcd for nany beautiful and interesting results and applications though the concept of generalizod inverse was first introduced by Moore in as cÃ¥riy as 1920 as follows :Definition fitoore) : Let A be a m x n matrix over the field of complex numbers. Then G is the generalized inverse of A if AG is the orthogonal projection operator projecting arbitrary vectors onto the column space of A and GA 15 the orthogonal projection operator projecting arbitrary vectors onto the column space of GMoore studied this concept and its properties in some details in 1935.In 1955, unaware of the earlier work of Moore, Penrose defined eneralized inverse of a matrix as follows :Definition (Penrose) : Let * be m x matrix over the field of complex tambers. ThenG is a generalited inverse of A if (1) AGA - A,(ii) GAG = G, (til) (AG)* AG and (iv) (GA)* = CA. In 1956 Penrese shored that this peneraiized inverse of a matrix is unique and discussed the properties and uses of this generalized inverso of a matrix in a systematic way.In 1956 Rado establised that the definition due to Penrose is equivalent to that of Moore. This unique generalizcd inverse is called Moore-Penrose inverse of a matrix.A similar notion was also used by delt and Duffin in 1953 under the name constrained inverse and by Altken with a different sysbolism in 1934.Unaware of the earlier work of Moore and contemporary work of Penrose, Rao in 1955, constructed a pscudoinverse of a singular matrix which does not satisfy all the conditions of Moore-Penrosc inverse and showed that it serves the same purpose as regular inverses of a nonsingular matrix in solving normal equations and also in commuting standard errors of least squares estÃ¬mators. In 1962 R2o defined a generalized inverse, formally, as follows, discussed its properties in greater details and its application to the problems of Mathematical Statistics.Definition (Rao) A be an m x matrix, Then an n x a matrix Let G is a g-inverse of A if x - Gy is a solution of the linear system Ax - y whenever it is consistent. Rao showed that the above definition is equivalent to the following definition which is also due to him

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