Generalized matrix inverses and the generalized Gauss-Markoff theorem

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 ofMathematics, Department ofGraduat