Inverse matrices

The inverse of a nonsingular matrix is in general difficult to compute. A standard approach is to compute the inverse is doing matrix elimination.

The following are some useful identities that are computationally advantageous in some special cases.

Some identities for matrix inversion

 * Rank one update formula (known as Sherman-Morrison formula): Iterative inversions of the form $$P(m+1)^{-1} = \left(P(m) + a_{m+1}\right)^{-1}$$ can be done efficiently via:

$$(P+aa^T)^{-1} = P^{-1} - \frac{1}{1+a^T P^{-1}a} (P^{-1}a)(P^{-1}a)^T$$
 * $$(I-C)^{-1} = I+C+C^2+C^3+\dots$$ (if the series converges ). This is also called Neumann series.
 * Matrix inversion lemma (known as Woodbury formula): $$(A+BC)^{-1} = A^{-1}-A^{-1} B (I+C A^{-1}B)^{-1}CA^{-1}$$, which is useful when $$A$$ is easy to invert and $$BC$$ is a low-rank matrix

Below are some identities from Carl D. Meyer (p122) that have some symmetry:
 * $$A(I-A)^{-1}=(I-A)^{-1}A$$
 * $$A(A+B)^{-1}B=B(A+B)^{-1}A = (A^{-1}+B^{-1})^{-1}$$