Abstract :

A solution $X$ to to the matrix equation $'AXA = A'$ is known as a generalized inverse (g-inverse) of $A$ and a solution $X$ to $'XAX = X'$ is called an outer-inverse of $A$. Given a square matrix $A$, we can choose $C$ disjoint with $A$ such that $A+C$ is nonsingular. The inverse of $A+C$ produces all g-inverses of $A$. By this Inverse Complemented Matrix (ICM) method we can characterize all g-inverses and outer-inverses of $A$. The ICM provides a better approach to study the Generalized Linear Model (GLM), $Y = X\beta + \epsilon, var (\epsilon) = \sigma^{2}V$, by decomposing the design matrix $X$ and variance matrix $V$ appropriately.