Many matrices contain useful block structure. The Schur complement formalizes block elimination: after partitioning a square matrix into blocks, the expression S = D - CA⁻¹B is the block that remains after eliminating C using A. This chapter uses that block-level row reduction to compute determinants and inverses.
Schur complement is ordinary row reduction performed at the block level.
The main point is not a new kind of operation, but the same elimination logic applied to structured blocks instead of individual entries.
It appears when the lower-left block C is eliminated using the invertible block A as a pivot. The same row-replacement logic used in ordinary elimination subtracts CA⁻¹ times the top block row from the bottom block row, leaving D - CA⁻¹B in the lower-right position.
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Because the formula uses A⁻¹, so A must be invertible to serve as the block pivot. If that pivot block is not invertible, the same elimination idea may have to start from the other diagonal block instead, just as ordinary row reduction may need a different pivot when the current pivot is zero.