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

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 8.1.1. 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.7.2.2., 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 8.1.2. A has both a left and right inverse iff A is square (n = m) Lemma 8.1.3. If A is square and has right or left inverse , then A is nonsingular and B is the inverse of A. Theorem 8.1.4. 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.8.1.1., L.8.1.3.: 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)

Notation and Results

Properties of Inverses

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

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

Orthogonal Matrices - II

Orthogonal Matrices - III

Permutation Matrices - I

Permutation Matrices - II

Results for Partitioned Matrices - I

Results for Partitioned Matrices - II

Algorithm for Obtaining Inverse of Nonsingular Triangular Matrix

Example – Algorithm and Check Results in EXCEL

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

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

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

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

Permutation Matrices and Partitioned Matrices - I

Permutation Matrices and Partitioned Matrices - II