6. (20 points) Let A = (ajj) E Rxn be symmetric positive definite. In the derivation of Cholesky decomposition algorithm for A, at step k (k > 1), we have computed the first k - 1 columns of L (i.e., L(:, 1 : k - 1) is computed) from the previous steps. In order to obtain the k-th column of L, we need to take the square root of akk - Zillkj- (a) Prove that k-1 akk kj = akk - OK-1AK-10k-1, where Ak-1 ( R(k-1)x(k-1) is the submatrix A(1 : k - 1, 1 : k - 1) and ak-1 6 Rk-1 is the submatrix A(1 : k - 1, k). (Hint: Consider the partitions of A and Ak.) (b) Use (a) to prove that akk - _;=1 12; > 0