Exam 1

1. Question

   For the matrix
   

   ${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{A}& =\left(\begin{array}{cccc}\hfill 16& \hfill 4& \hfill 16& \hfill -4\\ \hfill 4& \hfill 5& \hfill 6& \hfill -9\\ \hfill 16& \hfill 6& \hfill 33& \hfill -28\\ \hfill -4& \hfill -9& \hfill -28& \hfill 58\end{array}\right).\hfill \end{array}}$

   
   compute the matrix $L=({\mathit{\ell }}_{\mathit{ij}}{)}_{1\le i,j\le 4}$ from the Cholesky decomposition $A=L{L}^{\top }$.
   Which of the following statements are true?
   
   1. ${\mathit{\ell }}_{41}=-1$
   2. ${\mathit{\ell }}_{44}<4$
   3. ${\mathit{\ell }}_{22}=-1$
   4. ${\mathit{\ell }}_{11}>0$
   5. ${\mathit{\ell }}_{32}\le 1$
   
   

   Solution

   The decomposition yields
   

   ${\displaystyle \begin{array}{cc}\multicolumn{1}{c}{L}& =\left(\begin{array}{cccc}\hfill 4& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill 2& \hfill 0& \hfill 0\\ \hfill 4& \hfill 1& \hfill 4& \hfill 0\\ \hfill -1& \hfill -4& \hfill -5& \hfill 4\end{array}\right)\hfill \end{array}}$

   
   and hence:
   
   1. True. ${\mathit{\ell }}_{41}=-1$
   2. False. ${\mathit{\ell }}_{44}=4\nless 4$
   3. False. ${\mathit{\ell }}_{22}=2\ne -1$
   4. True. ${\mathit{\ell }}_{11}=4$
   5. True. ${\mathit{\ell }}_{32}=1$