# Posterior of factors in factor analysis

Based on the book by Bartholomew et al. (2011) and Equation (3.6) the posterior distribution of a factor analysis model is a multivariate normal $$$$N_q (\boldsymbol{\Lambda}^{T} (\boldsymbol{\Lambda} \boldsymbol{\Lambda} ^{T} + \boldsymbol{\Psi})^{-1}(\boldsymbol{x}-\boldsymbol{\mu}), (\boldsymbol{\Lambda}^{T}\boldsymbol{\Psi}^{-1}\boldsymbol{\Lambda}+\boldsymbol{I})^{-1} )$$$$ However, using the formulas for the conditional normal distributions, the respective covariance matrix is given by $$\boldsymbol\Sigma_{11} - \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \boldsymbol\Sigma_{21}$$ which in this case leads to $$$$\boldsymbol{I} - \boldsymbol\Lambda^{T} (\boldsymbol{\Lambda\Lambda^{T}+\Psi})^{-1} \boldsymbol\Lambda$$$$ I don't understand how is this the same with the covariance matrix indicated by Eq. (3.6).

$$(A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}$$
Identifying $$A = \mathbf{I}$$, $$U = \boldsymbol{\Lambda}^T$$, $$C = \boldsymbol{\Psi}^{-1}$$ and $$V = \boldsymbol{\Lambda}$$ yields the expression you seek.