1. Inverse Matrices
From: D.A. Harville, Matrix Algebra from a Statistician’s Perspective, Springer. Chapter 8

2. Basic Definitions and Results
Right Inverse of m×n A ≡ n×m R such that AR = Im
Left Inverse of m×n A ≡ n×m L such that LA = In
Lemma m×n A has a right inverse iff rank(A) = m (full row rank) and left inverse iff rank(A) = n (full column rank)
Proof: If rank(A) = m, by Th , there exists R s.t. AR = Im  A has right inverse
Conversely: If there exists R s.t. AR = Im, rank(A) ≥ rank(AR) = rank(Im) = m. Also rank(A) ≤ m  rank(A) = m
Corollary A has both a left and right inverse iff A is square (n = m)
Lemma If A is square and has right or left inverse , then A is nonsingular and B is the inverse of A.
Theorem A matrix is invertible iff it’s a square, nonsingular matrix. Any nonsingular matrix has unique inverse B which is its only right or left inverse.
Proof: L , L : If rank(A) = m = n, right inverse exists: AB = I  A ≡ invertible
For any C that is inverse of A, CA = I, and C = CI = CAB = IB = B ≡ unique (no other left/right inverse)

3. Notation and Results

4. Properties of Inverses

5. Multiplication (Pre and Post) by Full (Column and Row) Rank Matrix

6. Orthogonal Matrices - I
Square matrix A is an orthogonal matrix if A’A=AA’=I
Equivalently, if A is nonsingular and A-1 = A’

7. Orthogonal Matrices - II

8. Orthogonal Matrices - III

9. Permutation Matrices - I

10. Permutation Matrices - II

11. Results for Partitioned Matrices - I

12. Results for Partitioned Matrices - II

13. Algorithm for Obtaining Inverse of Nonsingular Triangular Matrix

14. Example – Algorithm and Check Results in EXCEL

15. Partitioned Matrices with 2 Row Blocks and 2 Column Blocks - I

16. Partitioned Matrices with 2 Row Blocks and 2 Column Blocks - II

17. Partitioned Matrices with 2 Row Blocks and 2 Column Blocks - III

18. Partitioned Matrices with 2 Row Blocks and 2 Column Blocks - IV
Finding rank(A) where A={Aij} i,j=1,2
Invert A11 or A22
Obtain the Schur Complement: A22-A21A11-1A12 or A11-A12A22-1A21
Find the rank of the Schur complement
Helpful when A11 or A22 is of full rank and of large order with known inverse matrix
Finding inverse of nonsingular matrix A: A-1 = B
Obtain B11, B12, B21, B22
Helpful when A11 or A22 is of full rank and of large order with known inverse matrix and/or if only B22 or B11 is of practical interest

19. Permutation Matrices and Partitioned Matrices - I

20. Permutation Matrices and Partitioned Matrices - II