Handbook of Linear Algebra, 2nd Edition

Chapter 27

Generalized Inverses of Matrices

Yimin Wei

Fudan University

Throughout this chapter, we will survey some basic results on the Moore-Penrose inverse, the Drazin inverse, and the outer inverse.

27.1 Notation and Terminology

Definitions:

Let F be a field (usually ℂ or ℝ), let A ∈ Fm × n, and let b ∈ Fm.

For integers m ≥ 1 and 1 ≤ k ≤ m,

$Q_{k,m}=\left\{\alpha :\alpha =\left({\alpha }_1,{\alpha }_2,...,{\alpha }_k\right),1\le {\alpha }_1<{\alpha }_2<\cdots <{\alpha }_k\le m,\text{and}{\alpha }_1,{\alpha }_2,...,{\alpha }_{k}are\text{integers}\right\}.$

For α ∈ Qk,m and β ∈ Ql,n, the submatrix A[α, β] is the k × l matrix whose (i, j)-th element is the (αi, βj) element of A; see Chapter 11 for additional information about submatrices. If |β| = n (or |α| = m), then the notation A[α, *] (or A[*, β] = A[α, β]) is used to denote A[α, β].

For α ∈ Qk, m, b[α] is the subvector of b, whose ...