Generalized Inverse Matrix

Definition

Let

• $A$ : Matrix of size $m\times n$
• $G$ : Matrix of size $n\times m$

If

$$AGA=A$$

Then we call $G$ the generalized inverse (or g-inverse) of $A$

Obtaining g-inverse matrix: General

Let $A$ : Matrix of size $m\times n$

1. Determine square submatrix $M$ of order $\mathrm{rank}(A)$ that is invertible
2. Create $n\times m$ zero matrix, and insert $(M^{-1}{)}^T$ where $M$ was taken from $A$

The result from the last step is $(A^{-1}{)}^T$, the transpose of generalized inverse of $A$ satisfying $A{A}^{-1}A=A$

Obtaining g-inverse matrix: Schur complement