Exam 1

1. Question

   For the matrix $$\begin{aligned} A &= \left( \begin{array}{rrrr} 16 & 4 & 16 & -4 \\ 4 & 5 & 6 & -9 \\ 16 & 6 & 33 & -28 \\ -4 & -9 & -28 & 58 \end{array} \right). \end{aligned}$$ 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. $\ell_{41} = -1$
   2. $\ell_{44} < 4$
   3. $\ell_{22} = -1$
   4. $\ell_{11} > 0$
   5. $\ell_{32} \le 1$
   
   

   Solution

   The decomposition yields $$\begin{aligned} L &= \left( \begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 4 & 1 & 4 & 0 \\ -1 & -4 & -5 & 4 \end{array} \right) \end{aligned}$$ and hence:

   
   
   1. True. $\ell_{41} = -1$
   2. False. $\ell_{44} = 4 \nless 4$
   3. False. $\ell_{22} = 2 \neq -1$
   4. True. $\ell_{11} = 4$
   5. True. $\ell_{32} = 1$