Date: Feb 25, 2013 2:05 PM
Author: fl
Subject: Question about Schur complement


I read a paper on matrix inversion using Schur complement. I do not get the idea though its description. Here is it:

For a compound matrix M in the Faddeev algorithm [4],

M =[ A B]
[-C D] (1)
where A, B, C, and D arematrices with size of (n×n), (n×l),
(m × n), and (m × l), respectively, the Schur complement,
D+C A^(-1) B, can be calculated provided that matrix A is nonsingular.
First, a row operation is performed to multiply the top
row by another matrix W and then to add the result to the
bottom row:
M =[ A B ]
[-C + WA D + WB] (2)

When the lower left-hand quadrant of matrix M is nullified,
the Schur complement appears in the lower right-hand
quadrant. Therefore,Wbehaves as a decomposition operator
and should be equal to

W = C A^(-1) (3)
such that
D + WB = D + C A^-1 B. (4)

By properly substituting matrices A, B, C, and D, the matrix
operation or a combination of operations can be executed via
the Schur complement, for example, as follows.

Matrix inversion:
D + C A^-1 B = A^-1 (5)
if B = C = I and D = 0.

I do not understand how it can get the inverse of A. In (5) left, it still substitutes A^-1 in order to get the right A^-1.

Could you tell me how to use Schur complement to get A^(-1)?