Compatibility condition under positive semi-definiteness

Determine whether the compatibility condition (I − S_{ββ} S_{ββ}^+) V_{βγ} = 0 necessarily holds when the block variance–covariance matrices V_{νν}, V_{ββ}, and V_{γγ} are positive semi-definite, where S_{ββ} := V_{ββ} − V_{βγ} V_{γγ}^+ V_{γβ} is the Schur complement and superscript + denotes the Moore–Penrose inverse; alternatively, construct counterexamples or identify additional assumptions under which the condition is guaranteed.

Background

To compare joint and sequential procedures in potentially singular (over-identified) settings, the paper employs Moore–Penrose inverses and block matrix decompositions. A key notion is the (V_{ββ}, V_{γγ})-compatibility of V_{βγ}, which involves two conditions. The first condition is known to hold for positive semi-definite matrices by a result of Gallier (2011).

The authors explicitly note uncertainty about whether the second compatibility condition follows from positive semi-definiteness alone, making its general validity an open technical question relevant to the decomposition used to derive optimal joint variances and the equivalence analysis.