v2009.01.01 - Convex Optimization

560 APPENDIX A. LINEAR ALGEBRA
A.4.0.0.5 Lemma. Rank of Schur-form block. [114] [112]
Matrix B ∈ R m×n has rankB≤ ρ if and only if there exist matrices A∈ S m
and C ∈ S n such that
[ ]
[ ]
A 0
A B
rank
0 T ≤ 2ρ and G =
C
B T ≽ 0 (1428)
C
⋄
Schur-form positive semidefiniteness alone implies rankA ≥ rankB and
rankC ≥ rankB . But, even in absence of semidefiniteness, we must always
have rankG ≥ rankA, rankB, rankC by fundamental linear algebra.
A.4.1
Determinant
[ ] A B
G =
B T C
(1429)
We consider again a matrix G partitioned like (1410), but not necessarily
positive (semi)definite, where A and C are symmetric.
When A is invertible,
When C is invertible,
detG = detA det(C − B T A −1 B) (1430)
detG = detC det(A − BC −1 B T ) (1431)
When B is full-rank and skinny, C = 0, and A ≽ 0, then [53,10.1.1]
detG ≠ 0 ⇔ A + BB T ≻ 0 (1432)
When B is a (column) vector, then for all C ∈ R and all A of dimension
compatible with G
detG = det(A)C − B T A T cofB (1433)
while for C ≠ 0
detG = C det(A − 1 C BBT ) (1434)
where A cof is the matrix of cofactors [287,4] corresponding to A .