For any multivariate Gaussian distribution which has $$ \mathbf{x} \sim \mathcal{N}(\mathbf{m},\mathbf{S}) $$ we have $$ \mathbf{m}_{a|b} = \mathbf{m}_a+\mathbf{S}_{ab}\mathbf{S}_{bb}^{-1}(\mathbf{x}_b-\mathbf{m}_b) $$ If $\mathbf{m} = m\overrightarrow{1}$, i.e., $\mathbf{m} = [m,m,...,m]^{T}$, do we have $$ \mathbf{m}_{a|b} = \mathbf{S}_{ab}\mathbf{S}_{bb}^{-1}\mathbf{x}_b? $$ For $m=0$, it is right. But I have no idea about the case $m \neq 0$.
This question is from the paper variational inference for gaussian process modulated poisson processes (see Equation 5 and helpful description above it). Thanks for any help.