LU_suggested.md

Linear Algebra Review

(Gauss Elimination) Suggested problems

No due date

1. Determine the lower (L) and upper (U) triangular matrices with LU-decomposition for the following matrices:

   a. $A=\left[ \begin{array}{cc} 1 & 3 \ 2 & 1 \end{array} \right]$

   a. $A=\left[ \begin{array}{cc} 1 & 1 \ 2 & 3 \end{array} \right]$

   a. $A=\left[ \begin{array}{cc} 1 & 1 \ 2 & -2 \end{array} \right]$

   b. $A=\left[ \begin{array}{ccc} 1 & 3 & 1 \ -4 & -9 & 2 \ 0 & 3 & 6\end{array} \right]$

   c. $A=\left[ \begin{array}{ccc} 1 & 3 & 1 \ -4 & -9 & 2 \ 0 & 3 & 6\end{array} \right]$

   d. $A=\left[ \begin{array}{ccc} 1 & 3 & -5 \ 1 & 4 & -8 \ -3 & -7 & 9\end{array} \right]$

   d. $A=\left[ \begin{array}{ccc} 1 & 2 & -1 \ 2 & 2 & 2 \ 1 & -1 & 2\end{array} \right]$

2. Calculate the determinant of A from 1a-g.

3. Determine the Cholesky factorization, C, of the following matrices, where

   $C_{ii}=\sqrt{a_{ii}-\sum_{k=1}^{i-1}C_{ki}^{2}}$

   $C_{ij}=\frac{a_{ij}-\sum_{k=1}^{i-1}C_{ki}C_{kj}}{C_{ii}}$.

   a. A=$\left[ \begin{array}{cc} 3 & 2 \ 2 & 1 \end{array} \right]$

   a. A=$\left[ \begin{array}{cc} 10 & 5 \ 5 & 20 \end{array} \right]$

   a. A=$\left[ \begin{array}{ccc} 10 & -10 & 20 \ -10 & 20 & 10 \ 20 & 10 & 30 \end{array} \right]$

   a. A=$\left[ \begin{array}{cccc} 21 & -1 & 0 & 0 \ -1 & 21 & -1 & 0 \ 0 & -1 & 21 & -1 \ 0 & 0 & -1 & 1 \end{array} \right]$

4. Verify that $C^{T}C=A$ for 3a-d