Linear rigidity and uniqueness of size-two positive semidefinite factorizations with zeros

For real size-two positive semidefinite factorizations
$M_{ij} = \operatorname{tr}(A_i B_j)$ of matrices $M$ of ordinary rank three,
the paper proves that linear (2-infinitesimal) rigidity, meaning that every
feasible straight-line first-order direction is a common scaling
$E_i = dA_i$, $F_j = -dB_j$, is equivalent to uniqueness of the factorization
up to the congruence action of $GL(2, \mathbb{R})$, also when $M$ has zero
entries. The proof assumes neither distinct factors, nor a generic zero
pattern, nor nonzero rows and columns. An integration lemma shows that
uniqueness implies linear rigidity for every such factorization. For the
converse, after zero rows and columns are removed, a zero entry is normalized
to a fixed orthogonal pair of rank-one factors, and any inequivalent
factorization is converted into an explicit feasible direction that is not a
scaling. Matrices with positive entries are covered by the earlier theorem of
Dawson, Hoşten, Kubjas and Metsälampi.