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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 \begin{equation} 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} ) \end{equation} 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 \begin{equation} \boldsymbol{I} - \boldsymbol\Lambda^{T} (\boldsymbol{\Lambda\Lambda^{T}+\Psi})^{-1} \boldsymbol\Lambda \end{equation} I don't understand how is this the same with the covariance matrix indicated by Eq. (3.6).

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The derivation relies on a result known as the matrix inversion lemma, or Woodbury matrix identity. From wikipedia:

$$(A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1}$$

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

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