I'd appreciate help finding the inverse of the upper-left triangular (partitioned) matrix
$$
\left[
\begin{array}{ll}
\mathbf{K} & \mathbf{P} \\
\mathbf{P}^T & \mathbf{0}
\end{array}
\right]
$$
This matrix occurs frequently in scattered data interpolation with radial basis functions. If it matters, $\mathbf{K}$ is a square matrix, $\mathbf{P}$ is generally not square and $\mathbf{0}$ is the zero matrix.
Best Answer
The answer above assumes the block $(1,1)$ of $A$ is nonsingular.
In the case of interpolation by radial basis functions, you do not have this necessarily.
So, you need a more general formula, given by Gansterer.
Look for it in a PDF text by Benzi and Golub on numerical solutions to saddle-point problems, easy to find on Google.