Application To Solving Linear Equations
The Schur complement arises naturally in solving a system of linear equations such as
where x, a are p-dimensional column vectors, y, b are q-dimensional column vectors, and A, B, C, D are as above. Multiplying the bottom equation by and then subtracting from the top equation one obtains
Thus if one can invert D as well as the Schur complement of D, one can solve for x, and then by using the equation one can solve for y. This reduces the problem of inverting a matrix to that of inverting a p×p matrix and a q×q matrix. In practice one needs D to be well-conditioned in order for this algorithm to be numerically accurate.
Read more about this topic: Schur Complement
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