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.

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    $\begingroup$ Does the blockwise inversion formula help? $\endgroup$ – Rahul Aug 5 '11 at 15:20
  • $\begingroup$ 'Just might -- as long as it does not involve taking the inverse of the zero matrix or of $\mathbf{P}$ and $\mathbf{P}^T$. Edit: looks like it does the trick -- thanks! $\endgroup$ – Olumide Aug 5 '11 at 15:40

To get this off the "unanswered" list, there is the formula (which I also used here):


for the block matrix $\mathbf{A}=\begin{pmatrix}\mathbf{E}&\mathbf{F}\\ \mathbf{G}&\mathbf{H}\end{pmatrix}$.

Applying this formula to your matrix yields



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.


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