Schur complement
Let A,B,C,D be matrices of sizes , , and respectively and suppose that is invertible. Let
so that is a matrix.
Then the Schur complement of the block of the matrix is the
matrix, . Analogously if is invertible then the Schur complement of the block of the matrix is the
matrix, .
In the first case, when is invertible, the Schur complement arises as the result of performing a partial Gaussian elimination by multiplying the matrix from the right with the lower triangular block matrix
,
where is the identity matrix and is the zero matrix
. Analogously, in the second case, we take the Schur complement by multiplying the matrix from the left with the lower triangular block matrix
see also:
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Wikipedia, http://en.wikipedia.org/wiki/Schur_complementSchur complement
Title | Schur complement |
---|---|
Canonical name | SchurComplement |
Date of creation | 2013-03-22 15:27:11 |
Last modified on | 2013-03-22 15:27:11 |
Owner | georgiosl (7242) |
Last modified by | georgiosl (7242) |
Numerical id | 8 |
Author | georgiosl (7242) |
Entry type | Definition |
Classification | msc 15A15 |
Related topic | BlockDeterminants |
Related topic | MatrixInversionLemma |