Proposition: Linear solvability does not imply positivity-compatible solvability

Constraints on a covariance can make the allowed correction set nonaffine. Therefore solving a linear Ward equation in an enlarged vector space need not solve the constrained problem.

Proof. Already for real symmetric matrices, start from ${\mu }_0=\mathrm{diag}(0,1)$. The allowed changes preserving ${\mu }_0+S\ge 0$ do not form a vector space: $S=\mathrm{diag}(1,0)$ is allowed, but $-S$ is not. A linear equation demanding the latter direction is solvable in the unrestricted symmetric-matrix space and not in the positivity-compatible set. The example proves the logical distinction without identifying this matrix problem with the native physical quotient.