Minors of the Moore - Penrose Inverse ∗

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 A[α′, β′]. For nonsingular A ∈ IRn×n, the Jacobi identity relates the minors of the inverse A−1 to those of A: detA−1[β, α] = (−1) ∑k i=1 αi+ ∑k i=1 βi detA[α ′, β′] detA for any α, β ∈ Qk,n. We generalize Jacobi’s identity to matrices A ∈ IRm×n r , 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, detA†[β, α] = 1 vol A ∑ (I,J)∈N (α, β) detAIJ ∂ ∂|Aαβ | |AIJ |, for any α ∈ Qk,m , β ∈ Qk,n , 1 ≤ k ≤ r, where vol A denotes the sum of squares of determinants of r × r submatrices of A. This represents the k× k minors of A† as a convex combination of the minors of A. The weights of this combination are, surprisingly, the same for all k. We apply our results to questions concerning the nonnegativity of principal minors of the Moore-Penrose inverse.