## Exam 1

1. #### Question

For the matrix \begin{aligned} A &= \left( \begin{array}{rrrr} 16 & -12 & -12 & -16 \\ -12 & 25 & 1 & -4 \\ -12 & 1 & 17 & 14 \\ -16 & -4 & 14 & 57 \end{array} \right). \end{aligned} compute the matrix $L = (\ell_{ij})_{1 \leq i,j \leq 4}$ from the Cholesky decomposition $A = L L^\top$.

Which of the following statements are true?

1. $\ell_{41} \ge -4$
2. $\ell_{33} \ge 2$
3. $\ell_{11} \le 4$
4. $\ell_{31} \ge -3$
5. $\ell_{32} \ge -2$

#### Solution

The decomposition yields \begin{aligned} L &= \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ -3 & 4 & 0 & 0 \\ -3 & -2 & 2 & 0 \\ -4 & -4 & -3 & 4 \end{array} \right) \end{aligned} and hence:

1. True. $\ell_{41} = -4$
2. True. $\ell_{33} = 2$
3. True. $\ell_{11} = 4$
4. True. $\ell_{31} = -3$
5. True. $\ell_{32} = -2$