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 AFm × n, and let bFm.

For integers m ≥ 1 and 1 ≤ km,

Qk,m={α:α=(α1,α2,...,αk),1α1<α2<<αkm, and α1,α2,...,αk are integers}.

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 ...

Get Handbook of Linear Algebra, 2nd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.