Related to a previous question: Suppose I want to invert a (sparse) matrix written in block form as \begin{array}{cccc} A_{11} & A_{12} & \ldots & A_{1n}\\ A_{21} & A_{22} & & \vdots\\ \vdots & & \ddots\\ A_{n1} & A_{n2} & \ldots & A_{nn} \end{array}
where all the $A_{i,j}$'s are diagonal. Is the best way to do this just repeated application of the partitioned matrix inverse formula? Also, do such matrices have a name?