symbolic - How to symbolically do matrix "Block Inversion"?

Consider a block (partitioned) matrix

matrix = ArrayFlatten[{{a, b}, {c, d}}]

where, a, b, c and d are each matrices themselves. Say, for example,

a = {{a11, a12}, {a21, a22}}
b = {{b11, b12}, {b21, b22}}
c = {{0, 0}, {0, 0}}

d = {{d11, d12}, {d21, d22}}

How can you find the block inverse of this matrix? A desired solution is, using the example above

{{Inverse[a] , -Inverse[a].b.Inverse[d]},{0,Inverse[d]}}

which is easily verified using

Simplify[Inverse[ArrayFlatten[{{a, b}, {c, d}}]] == 
ArrayFlatten[{{Inverse[a], -Inverse[a].b.Inverse[d]}, {0, 
 Inverse[d]}}]]

which yields True.

How can you solve the block inverse problem for arbitrary submatrices, and for block matrices of larger sizes (i.e. 3x3, 4x4, etc)?

Answer

Mathematica does not support this directly. You can do things of this sort using an external package called NCAlgebra.

http://math.ucsd.edu/~ncalg/

The relevant documentation may be found at

http://math.ucsd.edu/~ncalg/DOWNLOAD2010/DOCUMENTATION/html/NCBIGDOCch4.html#x8-510004.4

In particular have a look at "4.4.8 NCLDUDecomposition[aMatrix, Options]"

Using this package, you would find the block inverse of the example matrix using:

c=0;    

inverse = NCInverse[matrix]  
(* Out[] = {{inv[a], -inv[a] ** b ** inv[d]}, {0, inv[d]}} *)

Here inv[a] represents the general inverse of the a block of the matrix and the ** represents non-commutative (i.e. matrix) multiplication. This approach works for larger (3x3, 4x4, etc) square block matrices as well.