## Exam 1

1. #### Question

For the matrix
 $\begin{array}{cc}\multicolumn{1}{c}{A}& =\left(\begin{array}{cccc}\hfill 16& \hfill -12& \hfill -12& \hfill -16\\ \hfill -12& \hfill 25& \hfill 1& \hfill -4\\ \hfill -12& \hfill 1& \hfill 17& \hfill 14\\ \hfill -16& \hfill -4& \hfill 14& \hfill 57\end{array}\right).\hfill \end{array}$

compute the matrix $L=\left({\ell }_{\mathrm{ij}}{\right)}_{1\le i,j\le 4}$ 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
 $\begin{array}{cc}\multicolumn{1}{c}{L}& =\left(\begin{array}{cccc}\hfill 4& \hfill 0& \hfill 0& \hfill 0\\ \hfill -3& \hfill 4& \hfill 0& \hfill 0\\ \hfill -3& \hfill -2& \hfill 2& \hfill 0\\ \hfill -4& \hfill -4& \hfill -3& \hfill 4\end{array}\right)\hfill \end{array}$

and hence:
1. True. ${\ell }_{41}=-4$
2. True. ${\ell }_{33}=2$
3. True. ${\ell }_{11}=4$
4. True. ${\ell }_{31}=-3$
5. True. ${\ell }_{32}=-2$