We demonstrate that subject to certain regularity conditions any invertible matrix whose inverse is subordinate to a chordal graph G may be inverted via a simple formula involving only the inverses of its principal submatrices corresponding to the maximal cliques and minimal vertex separators of the graph G. The resulting formula is reminiscent of known formulae for the determinant and inertia of matrices whose inverses are subordinate to a chordal graph. © 1998 Elsevier Science Inc. All rights reserved.
Johnson, C. R., & Lundquist, M. (1998). Local inversion of matrices with sparse inverses. Linear Algebra and Its Applications, 277(1–3), 33–39. https://doi.org/10.1016/S0024-3795(97)10071-4