A Completely New Method For Computing Generalized Inverses Based On Their Space Mapping Properties And Their Applications To Linear Model

Generalized inverse is the important invention of matrix theory in the twentieth century. It is applied in many fields of probability and Statistic analysis, especially in the estimate of parameters of linear models and multi-analysis. So the study on generalized inverses is significant. In this paper, the author make a deep study on generalized inverses, consider generalized inverses from a completely new angle, and establish a completely new method for computing generalized inverses. Major results are as follows: 1. firstly the author study the three generalized inverses based on {2}-inverse, that is {2,3}-inverse, {2,4}-inverse and {2,3,4}-inverse. From the viewpoint of algebra, the author give their some properties and algebraic constructions . 2. The researches on generalized inverse are from a point of algebra for a long time., and various methods for computing generalized inverses have been established by means of complicated matrix operations such as matrix multiplication, iteration, inversion,etc. In this paper, the author study generalized inverses from a completely new angle, for the first time consider generalized inverses as space mapping operators instead of only a matrix of algebra, and present the space mapping properties and characterization of the {1}-inverse, {2}-inverse and {1,2}-inverse which also reveals the essential defferences of {1}-inverse and {2}-inverse. For some existing properties of {1}-inverse and {2}-inverse, here the author give them another proof based on the space mapping properties of {1}-inverse and {2}-inverse. 3. Based on the space mapping properties of generalized inverses, completely new methods for computing A ?, AL ?, M, A (1,2)and AL (1,,M2) are established. Compared with the earlier method for computing generalized inverses , our method is much more simple and clear. 4. the author give some important applications of generalized inverses to linear models. We can apply the above new methods for computing generalized inverses in linear models, especially in case the data of design matrix X and error covariance matrix Σis complex, our new methods for computing generalized inverses appear more simple and clear.