INVERS MOORE PENROSE DIPERUMUM DAN INVERS ?-MOORE PENROSE DIPERUMUM PADA RING DENGAN ELEMEN SATUAN YANG DILENGKAPI INVOLUSI; GENERALIZED MOORE PENROSE INVERSE AND GENERALIZED ?-MOORE PENROSE INVERSE IN A RING WITH UNIT ELEMENT AND EQUIPPED BY INVOLUTION

SRRM, Titi Udjiani, Sri Wahyuni

2016 | Tesis | FMIPA

The Moore Penrose inverse in rings R with involution "?" is built by normalized inner inverse. This dissertation obtains the definition of the generalized Moore Penrose inverse in rings which is a generalization of the definition of the Moore Penrose inverse. The idea for generalizing the Moore Penrose inverse is motivated by a finding which explains that the normalized inner inverse can be obtained by the any inverse. Generalization is contructed by ignoring the normalized of the inner inverse. Because of not each element in R has an inner inverse, so not each element in R has the Moore Penrose generalized inverse. The necessary and sufficient condition of an element has the generalized Moore Penrose inverse is constructed with the first determining of the necessary and sufficient conditions of an element having the inner inverse. Then the obtained result was developed to obtain the necessary and sufficient conditions that an element in rings has the generalized Moore Penrose inverse. Using the properties of the generalized Moore Penrose inverse which obtained directly from the definition of generalized Moore Penrose inverse, can be constructed the properties of the elements in the R which associated with generalized Moore Penrose inverse. Those are symmetrice element, generalized symmetrice element, normal element , generalized normal element, star dagger element , Enhancer promoter element and a partial isometry element. In this research, the involution "?" can be generalized into involution "?u", u2R, u2 = u and u? = 1. Motivated by this invention, then can be constructed the definition of the generalized ? -Moore Penrose inverse which is generalization of the generalized Moore Penrose inverse. The final part of this dissertation describes that vector regression parameter estimators in a linear regression model can be built by the generalized Moore Penrose inverse

Kata Kunci : regular element; inner inverse; Moore Penrose inverse.