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106 lines
2.6 KiB
TeX
106 lines
2.6 KiB
TeX
\section*{Aufgabe 1}
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\paragraph{Gegeben:}
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\[A = \begin{pmatrix}
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2 & 3 & -1\\
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-6 & -5 & 0\\
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2 & -5 & 6
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\end{pmatrix},\;\;\; b = \begin{pmatrix}20\\-41\\-15\end{pmatrix}\]
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\paragraph{LR-Zerlegung:}
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\begin{align}
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&&A^{(0)} &= \begin{gmatrix}[p]
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2 & 3 & -1\\
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-6 & -5 & 0\\
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2 & -5 & 6
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\rowops
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\swap{0}{1}
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\end{gmatrix}\\
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P^{(1)} &= \begin{pmatrix}
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0 & 1 & 0\\
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1 & 0 & 0\\
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0 & 0 & 1
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\end{pmatrix}
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&A^{(1)} &=
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\begin{gmatrix}[p]
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-6 & -5 & 0\\
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2 & 3 & -1\\
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2 & -5 & 6
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\rowops
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\add[\cdot \frac{1}{3}]{0}{1}
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\add[\cdot \frac{1}{3}]{0}{2}
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\end{gmatrix}\\
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L^{(2)} &=\begin{pmatrix}
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1 & 0 & 0\\
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\nicefrac{1}{3} & 1 & 0\\
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\nicefrac{1}{3} & 0 & 1
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\end{pmatrix},
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& A^{(2)} &= \begin{gmatrix}[p]
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-6 & -5 & 0\\
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0 & \frac{4}{3} & -1\\
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0 & -\frac{20}{3} & 6
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\rowops
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\swap{1}{2}
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\end{gmatrix}\\
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P^{(3)} &= \begin{pmatrix}
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1 & 0 & 0\\
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0 & 0 & 1\\
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0 & 1 & 0
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\end{pmatrix},
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& A^{(3)} &= \begin{gmatrix}[p]
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-6 & -5 & 0\\
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0 & -\frac{20}{3} & 6\\
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0 & \frac{4}{3} & -1
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\rowops
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\add[\cdot \frac{1}{5}]{1}{2}
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\end{gmatrix}\\
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L^{(4)} &= \begin{pmatrix}
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1 & 0 & 0\\
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0 & 1 & 0\\
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0 & \nicefrac{1}{5} & 1
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\end{pmatrix},
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& A^{(4)} &= \begin{gmatrix}[p]
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-6 & -5 & 0\\
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0 & -\frac{20}{3} & 6\\
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0 & 0 & \nicefrac{1}{5}
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\end{gmatrix} =:R
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\end{align}
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Es gilt nun:
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\begin{align}
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P :&= P^{(3)} \cdot P^{(1)}\\
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&= \begin{pmatrix}
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1 & 0 & 0\\
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0 & 0 & 1\\
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0 & 1 & 0
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\end{pmatrix} \cdot \begin{pmatrix}
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0 & 1 & 0\\
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1 & 0 & 0\\
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0 & 0 & 1
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\end{pmatrix} \\
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&=
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\begin{pmatrix}
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0 & 1 & 0\\
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0 & 0 & 1\\
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1 & 0 & 0
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\end{pmatrix}\\
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L^{(4)} \cdot P^{(3)} \cdot L^{(2)} \cdot P^{(1)} \cdot A &= R\\
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L^{-1} &= L^{(4)} \cdot \hat{L_1}\\
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\hat{L_1} &= P^{(3)} \cdot L^{(2)} \cdot (P^{(3)})^{-1}\\
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&= P^{(3)} \cdot L^{(2)} \cdot P^{(3)}\\
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&= \begin{pmatrix}
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1 & 0 & 0\\
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\nicefrac{1}{3} & 1 & 0\\
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\nicefrac{1}{3} & 0 & 1
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\end{pmatrix}\\
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L &= (L^{(4)} \cdot \hat{L_1})^{-1}\\
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&= \begin{pmatrix}
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1 & 0 & 0\\
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-\frac{1}{3} & 1 & 0\\
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-\frac{1}{3} & -\frac{1}{5} & 1
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\end{pmatrix}
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\end{align}
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Überprüfung mit \href{http://www.wolframalpha.com/input/?i=%7B%7B1%2C+0%2C+0%7D%2C+%7B-1%2F3%2C+1%2C+0%7D%2C+%7B-1%2F3%2C+-1%2F5%2C+1%7D%7D*%7B%7B-6%2C-5%2C0%7D%2C%7B0%2C-20%2F3%2C6%7D%2C%7B0%2C0%2C1%2F5%7D%7D}{Wolfram|Alpha}.
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