Exam 1

  1. Question

    For the matrix A=(1612121612251412117141641457). \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=(ij)1i,j4L = (\ell_{ij})_{1 \leq i,j \leq 4} from the Cholesky decomposition A=LLA = L L^\top.

    Which of the following statements are true?


    1. 414\ell_{41} \ge -4
    2. 332\ell_{33} \ge 2
    3. 114\ell_{11} \le 4
    4. 313\ell_{31} \ge -3
    5. 322\ell_{32} \ge -2

    Solution

    The decomposition yields L=(4000340032204434) \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. 41=4\ell_{41} = -4
    2. True. 33=2\ell_{33} = 2
    3. True. 11=4\ell_{11} = 4
    4. True. 31=3\ell_{31} = -3
    5. True. 32=2\ell_{32} = -2