Generalized Inverse Matrices From D A Harville Matrix

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

Introduction

Obtaining Generalized Inverses for Various Cases - I

Obtaining Generalized Inverses for Various Cases - II

Obtaining Generalized Inverses for Various Cases - III

Algorithm to Obtain Generalized Inverse G for Matrix A 1. Obtain the rank of the matrix (maximum number of linearly independent rows/columns). Let rank = r 2. Identify r linearly independent rows: i 1, …, ir and r linearly independent columns j 1, …, jr 3. Obtain the nonsingular submatrix for those rows and columns: B 11 4. Obtain the inverse of B 11 -1 5. Place element (s, t) from B 11 -1 in cell (js , it) of G (s=1, …, r; t=1, …, r) 6. Set all other elements of G to 0 (Elements not in rows j 1, …, jr or columns i 1, …, ir )

Example

Generalized Inverses for Symmetric Matrices

G-Inverses Based on a Particular G-Inverse

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

Some Properties of G-Inverses - I

Some Properties of G-Inverses - II

Invariance Among G-Inverses

Condition for Consistency of a Linear System

Ranks of G-Inverses of Partitioned Matrices - I

Ranks of G-Inverses of Partitioned Matrices - II

Ranks of G-Inverses of Partitioned Matrices - III

Extensions of Systems AX=B to AXC=B