

\begin{question}
For the matrix
\begin{eqnarray*}
  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{eqnarray*}
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?
\begin{answerlist}
  \item $\ell_{41} = -1$
  \item $\ell_{44} < 4$
  \item $\ell_{22} = -1$
  \item $\ell_{11} > 0$
  \item $\ell_{32} \le 1$
\end{answerlist}
\end{question}

\begin{solution}
The decomposition yields
\begin{eqnarray*}
  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{eqnarray*}
and hence:
\begin{answerlist}
  \item True. $\ell_{41} = -1$
  \item False. $\ell_{44} = 4 \not< 4$
  \item False. $\ell_{22} = 2 \not= -1$
  \item True. $\ell_{11} = 4$
  \item True. $\ell_{32} = 1$
\end{answerlist}
\end{solution}

%% META-INFORMATION
%% \extype{mchoice}
%% \exsolution{10011}
%% \exname{Cholesky decomposition}
