I wonder how to express the determinant of a block covariance matrix. For example, I have a covariance matrix


$\Sigma=\left[
 \begin{array}{cc}
 \Sigma\_1 & \Sigma\_{12} \\
 \Sigma\_{21} & \Sigma\_2 \\
 \end{array}
\right]$


where $\Sigma\_1\in\mathbb{R}^{n\_1}\times\mathbb{R}^{n\_1}$, $\Sigma\_2\in\mathbb{R}^{n\_2}\times\mathbb{R}^{n\_2}$ and hence $\Sigma\_{12}=\Sigma\_{21}^\intercal\in\mathbb{R}^{n\_1}\times\mathbb{R}^{n\_2}$.


Can I express the denterminant of $\Sigma$ in terms of the 3 blocks? That is:


$|\Sigma|=f(\Sigma\_1,\Sigma\_2,\Sigma\_{12})$


I guess it is somehow related to Schur Complement, I am still studying..



Determinant of block covariance matrix
A well-known block determinant formula: if $\Sigma\_1$ is invertible,
$$\det \Sigma = \det(\Sigma\_1) \det(\Sigma\_2 - \Sigma\_{21} \Sigma\_1^{-1} \Sigma\_{12})$$ (and a similar formula if $\Sigma\_2$ is invertible).


