## Abstract

Let Q_{k},_{n} = {α = (α_{1},..., α_{k}): 1 ≤ α_{1} < ⋯ < α_{k} ≤ n} denote the strictly increasing sequences of k elements from 1,...,n. For α, β ∈ Q_{k},_{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 ∈ R^{n×n} the Jacobi identity relates the minors of the inverse A^{-1} to those of A: det A^{-1}lsqbβ, αrsqb = (-1)^{Σki=1αi+Σ ki=1βi} detAlsqbα′, βrsqb det A for any α, β ∈ Q_{k},_{n}. We generalize the Jacobi identity to matrices A ∈ R^{m×n}_{r}, expressing the minors of the Moore-Penrose inverse A^{†} in terms of the minors of the maximal nonsingular submatrices A_{IJ} of A. In our notation, det A^{†}lsqbβ, αrsqb = 1 vol^{2}A ∑ (I,J)∈N(α, β) det A_{IJ} ∂ ∂|A_{αβ}||A_{IJ}| for any α ∈ Q_{k},_{m}, β ∈ Q_{k},_{n}, 1 ≤ k ≤ r, where vol^{2} 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 language | English (US) |
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Pages (from-to) | 191-207 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 195 |

Issue number | C |

DOIs | |

State | Published - Dec 1993 |

## All Science Journal Classification (ASJC) codes

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