In this thesis we present the generalization of the Moore-Penrose pseudo-inverse in the sense that it satisfies the following conditions. Let x be an m × n matrix of rank r , and let u and v be symmetric positive semi-definite matrices of order m and n and rank s and t respectively, such that s.t ≥ r , and column space of x ⊂ column space of u row space of x⊂ row space of v.
Then x≠ is called the generalized inverse of x with respect to u and v if and only if it satisfies :
(i) xx≠x = x
(ii) x≠xx≠= x≠
(iii) (xx≠)’ = u⁺xx≠u
(iv) (x≠x)' = v⁺x≠xv ,
where U⁺ and V⁺ are the Moore-Penrose pseudo-inverses of U and V respectively. We further use this result to generalize the fundamental Gauss-Markoff theorem for linear estimation, and we also use it in the minimum mean square error estimation of the general model y = Xβ + ε , that is, we allow the covariance matrix of y to be symmetric positive semi-definite. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/33672 |
| Date | January 1971 |
| Creators | Ang , Siow-Leong |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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