Superscript notation inverse block matrix.

Question

Is it conventional to use superscript notation to denote the inverse of a block matrix. The context is the quadratic expression in the exponential of a multivariate Gaussian probability density distribution:

$$-\frac{1}{2}({\bf X - \mu})^T\, \Sigma^{-1}\, ({\bf X - \mu})$$

The covariance matrix $\Sigma$ is a block matrix:

$$\Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix}$$

The question is how accepted or conventional is it to express

$$\Sigma^{\color{red}{-1}} = \begin{bmatrix} \Sigma^{\color{red}{11}} & \Sigma^{\color{red}{12}} \\ \Sigma^{\color{red}{21}} & \Sigma^{\color{red}{22}} \end{bmatrix}$$

?

Equivalently, I guess I could be asking about common notation for the precision matrix.

Answer 1

I have never seen superscripts to denote blocks (or elements) of the inverse matrix. This does not indicate that it has never been used.