List of Taylor polyhedra: Difference between revisions

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en>Yobot
m WP:CHECKWIKI error fixes using AWB (8976)
en>David Eppstein
let's call the further reading section references, but complain about the lack of footnotes, rather than tagging this as completely unreferenced
 
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{{Multiple issues|orphan = September 2012|lead missing = September 2012|context = August 2012}}
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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>
 
Here ''I''<sub>''p''</sub> denotes a ''p''&times;''p'' [[identity matrix]]. After multiplication with the matrix ''L'' the Schur complement appears in the upper ''p''&times;''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]]

Latest revision as of 22:51, 20 December 2014

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