1. Generalized Inverse Matrices
From: D.A. Harville, Matrix Algebra from a Statistician’s Perspective, Springer. Chapter 9

2. Introduction

3. Obtaining Generalized Inverses for Various Cases - I

4. Obtaining Generalized Inverses for Various Cases - II

5. Obtaining Generalized Inverses for Various Cases - III

6. Algorithm to Obtain Generalized Inverse G for Matrix A
Obtain the rank of the matrix (maximum number of linearly independent rows/columns). Let rank = r
Identify r linearly independent rows: i1,…,ir and r linearly independent columns j1,…,jr
Obtain the nonsingular submatrix for those rows and columns: B11
Obtain the inverse of B11 B11-1
Place element (s,t) from B11-1 in cell (js , it) of G (s=1,…,r; t=1,…,r)
Set all other elements of G to 0 (Elements not in rows i1,…,ir or columns j1,…,jr )

7. Generalized Inverses for Symmetric Matrices

8. G-Inverses Based on a Particular G-Inverse

9. G-Inverses of Full Column Rank or Full Row Rank Matrices

10. Some Properties of G-Inverses - I

11. Some Properties of G-Inverses - II

12. Invariance Among G-Inverses

13. Condition for Consistency of a Linear System

14. Ranks of G-Inverses of Partitioned Matrices - I

15. Ranks of G-Inverses of Partitioned Matrices - II

16. Ranks of G-Inverses of Partitioned Matrices - III

17. Extensions of Systems AX=B to AXC=B