(Solved): 2. The LU decomposition decomposes a matrix A as the product of a lower triangular matrix L and an ...

2. The LU decomposition decomposes a matrix A as the product of a lower triangular matrix L and an upper triangular matrix U. If A is symmetric positive definite, U can be taken as LT and if we further requires Li> 0 (diagonal entries of L), this decomposition is unique and is called the Cholesky factorization of matrix A (see for example the book by Trefethen and Bau). We have the following theorem (which we will use directly) Theorem 1. If A € Rnxn symmetric positive definite, then there exists a unique lower triangular LE Rnxn with positive diagonal entries such that A = LLT. One immediately note that the entries of G can be computed one by one, from left to right (column by column), and from up to down. We will not go into the details, but simply cite the algorithm: for j = 1:n end v(j : n) = A(j : n, j) for k=1: j-1 v(j : n) = v(j : n) — L(j, k)L(j : n, k) end L(j : n, j) = v(j : n)/?v(j) Now, suppose A € Rnxn is symmetric and positive definite. Consider the following iteration: Ao = A For k = 0, 1, 2, ... (b) Show that if Ak LKL (Cholesky) Ak+1 = L Lk. Here L is lower triangular with positive diagonal elements. (a) Show that A is symmetric and positive definite (so that the iteration is well defined) and similar to A. [5 marks] a b +-(88) A = b C has eigenvalues A? A? > 0, then the A converges to diag(A?, A2). [Hint: Carry on one step in the iteration manually.] [5 marks] a> c