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
−1lsqbβ, αrsqb = (−1)
Σ
k
i=1
α
i
+Σ
k
i=1
β
i
det
Alsqbα′, 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.