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,

${Q}_{k,m}=\left\{\alpha :\alpha =\left({\alpha}_{1},{\alpha}_{2},...,{\alpha}_{k}\right),1\le {\alpha}_{1}<{\alpha}_{2}<\cdots <{\alpha}_{k}\le m,\text{}\text{and}\text{}{\alpha}_{1},{\alpha}_{2},...,{\alpha}_{k}\text{}are\text{}\text{integers}\right\}.$

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 ...

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