LU decomposição de $$$\left[\begin{array}{ccc}1 & 1 & -2\\1 & 3 & -1\\2 & 1 & -5\end{array}\right]$$$

Encontre a decomposição LU de $$$\left[\begin{array}{ccc}1 & 1 & -2\\1 & 3 & -1\\2 & 1 & -5\end{array}\right]$$$.

Comece pela matriz identidade $$$L = \left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$.

Subtraia a linha $$$1$$$ da linha $$$2$$$: $$$R_{2} = R_{2} - R_{1}$$$.

$$$\left[\begin{array}{ccc}1 & 1 & -2\\0 & 2 & 1\\2 & 1 & -5\end{array}\right]$$$

Escreva o coeficiente $$$1$$$ na matriz $$$L$$$ na linha $$$2$$$, coluna $$$1$$$:

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\0 & 0 & 1\end{array}\right]$$$

Subtraia a linha $$$1$$$ multiplicada por $$$2$$$ da linha $$$3$$$: $$$R_{3} = R_{3} - 2 R_{1}$$$.

$$$\left[\begin{array}{ccc}1 & 1 & -2\\0 & 2 & 1\\0 & -1 & -1\end{array}\right]$$$

Escreva o coeficiente $$$2$$$ na matriz $$$L$$$ na linha $$$3$$$, coluna $$$1$$$:

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\2 & 0 & 1\end{array}\right]$$$

Adicione a linha $$$2$$$ multiplicada por $$$\frac{1}{2}$$$ à linha $$$3$$$: $$$R_{3} = R_{3} + \frac{R_{2}}{2}$$$.

$$$\left[\begin{array}{ccc}1 & 1 & -2\\0 & 2 & 1\\0 & 0 & - \frac{1}{2}\end{array}\right]$$$

Escreva o coeficiente $$$- \frac{1}{2}$$$ na matriz $$$L$$$ na linha $$$3$$$, coluna $$$2$$$:

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\2 & - \frac{1}{2} & 1\end{array}\right]$$$

A matriz obtida é a matriz $$$U$$$.

$$$L = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\2 & - \frac{1}{2} & 1\end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 0\\1 & 1 & 0\\2 & -0.5 & 1\end{array}\right]$$$A

$$$U = \left[\begin{array}{ccc}1 & 1 & -2\\0 & 2 & 1\\0 & 0 & - \frac{1}{2}\end{array}\right] = \left[\begin{array}{ccc}1 & 1 & -2\\0 & 2 & 1\\0 & 0 & -0.5\end{array}\right]$$$A