3.4 LU Factorization

1. Use the LU factorization of $A$ to solve the linear system $A\mathbf{x}=\mathbf{b}$.

$$A=LU=\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ 2& 1& 1\end{array}\right]\left[\begin{array}{ccc}-7& 2& 2\\ 0& 1& 3\\ 0& 0& 5\end{array}\right]=\left[\begin{array}{ccc}-7& 2& 2\\ 21& -5& -3\\ -14& 5& 12\end{array}\right],\mathbf{b}=\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]$$ $$x_1=3$$ $$-3x_1+x_2=-1$$ $$2x_1+x_2+x_3=4$$ $$x_2=8$$ $$x_3=-10$$

2. Find the LU factorization of the following matrices:

$$A=\left[\begin{array}{ll}2& -1\\ 8& -1\end{array}\right]$$ $$\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right]$$ $$B=\left[\begin{array}{ccc}2& -1& 0\\ 4& 1& 1\\ -2& 4& -1\end{array}\right]$$ $$\left[\begin{array}{ccc}2& -1& 0\\ 0& 3& 1\\ 0& 0& -2\end{array}\right]$$

3. Find the LDU factorization of

$$B=\left[\begin{array}{ccc}2& -1& 0\\ 4& 1& 1\\ -2& 4& -1\end{array}\right]$$ L = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ -1 & 1 & 1 \end{bmatrix} $$ $$ D = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -2 \end{bmatrix} $$ $$ U = \begin{bmatrix} 1 & -\frac{1}{2} & 0 \\ 0 & 1 & \frac{1}{3} \\ 0 & 0 & 1 \end{bmatrix} $$