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fl
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90
Registered:
10/8/05


Question about Schur complement
Posted:
Feb 25, 2013 2:05 PM


Hi,
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 lefthand quadrant of matrix M is nullified, the Schur complement appears in the lower righthand 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)?
Thanks



