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 ,
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)?