Minors of the Moore-Penrose inverse

Jianming Miao, Adi Ben-Israel

Research output: Contribution to journalArticlepeer-review

17 Scopus citations

Abstract

Let Qk,n = {α = (α1,..., αk): 1 ≤ α1 < ⋯ < αk ≤ n} denote the strictly increasing sequences of k elements from 1,...,n. For α, β ∈ Qk,n we denote by A[α, β] the submatrix of A with rows indexed by α, columns by β. The submatrix obtained by deleting the α-rows and β-columns is denoted by Alsqbα′, β′rsqb. For nonsingular A ∈ Rn×n the Jacobi identity relates the minors of the inverse A-1 to those of A: det A-1lsqbβ, αrsqb = (-1)Σki=1αiki=1βi detAlsqbα′, βrsqb det A for any α, β ∈ Qk,n. We generalize the Jacobi identity to matrices A ∈ Rm×nr, expressing the minors of the Moore-Penrose inverse A in terms of the minors of the maximal nonsingular submatrices AIJ of A. In our notation, det Alsqbβ, αrsqb = 1 vol2A ∑ (I,J)∈N(α, β) det AIJ ∂ ∂|Aαβ||AIJ| for any α ∈ Qk,m, β ∈ Qk,n, 1 ≤ k ≤ r, where vol2 A denotes the sum of squares of determinants of r×r submatrices of A. We apply our results to questions concerning the nonnegativity of principal minors of the Moore-Penrose inverse.

Original languageEnglish (US)
Pages (from-to)191-207
Number of pages17
JournalLinear Algebra and Its Applications
Volume195
Issue numberC
DOIs
StatePublished - Dec 1993

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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