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| | {{Multiple issues|orphan = September 2012|lead missing = September 2012|context = August 2012}} |
| | |
| | The following identity holds for a [[Schur complement]] of a square [[matrix (mathematics)|matrix]]: |
| | |
| | The Schur complement arises as the result of performing a block [[Gaussian elimination]] by multiplying the matrix ''M'' from the right with the "block lower triangular" matrix |
| | |
| | :<math>L=\left[\begin{matrix} I_p & 0 \\ -D^{-1}C & I_q \end{matrix}\right].</math> |
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| | Here ''I''<sub>''p''</sub> denotes a ''p''×''p'' [[identity matrix]]. After multiplication with the matrix ''L'' the Schur complement appears in the upper ''p''×''p'' block. The product matrix is |
| | |
| | :<math> |
| | \begin{align} |
| | ML &= \left[\begin{matrix} A & B \\ C & D \end{matrix}\right]\left[\begin{matrix} I_p & 0 \\ -D^{-1}C & I_q \end{matrix}\right] = \left[\begin{matrix} A-BD^{-1}C & B \\ 0 & D \end{matrix}\right] \\ |
| | &= \left[\begin{matrix} I_p & BD^{-1} \\ 0 & I_q \end{matrix}\right] \left[\begin{matrix} A-BD^{-1}C & 0 \\ 0 & D \end{matrix}\right]. |
| | \end{align} |
| | </math> |
| | That is, we have shown that |
| | |
| | :<math> |
| | \begin{align} |
| | \left[\begin{matrix} A & B \\ C & D \end{matrix}\right] &= \left[\begin{matrix} I_p & BD^{-1} \\ 0 & I_q \end{matrix}\right] \left[\begin{matrix} A-BD^{-1}C & 0 \\ 0 & D \end{matrix}\right] |
| | \left[ \begin{matrix} I_p & 0 \\ D^{-1}C & I_q \end{matrix}\right], |
| | \end{align} |
| | </math> |
| | |
| | [[Category:Gaussian function]] |
Revision as of 13:16, 15 March 2013
Template:Multiple issues
The following identity holds for a Schur complement of a square matrix:
The Schur complement arises as the result of performing a block Gaussian elimination by multiplying the matrix M from the right with the "block lower triangular" matrix
Here Ip denotes a p×p identity matrix. After multiplication with the matrix L the Schur complement appears in the upper p×p block. The product matrix is
That is, we have shown that