ALU The Cholesky Factorization Proof and Bordered Algorithm SPD Matrix Worksheet

Need homework help for Cholesky factorization proof. Please see the attached pdf for the question requirements.

Question #1: This problem covers Cholesky factorization
(https://en.wikipedia.org/wiki/Cholesky_decomposition ).
(a) Propose a bordered algorithm for computing the Cholesky factorization of a SPD matrix
A inspired by the bordered LU factorization algorithm. Show your derivation that
justifies the algorithm. You don’t need to use the formalisms in 5.5.1, but you do need to
derive the algorithm. You may want to do so by partitioning:
Note: you can likely find such an algorithm online or in the literature and that is permitted. (Although, you’ll learn more if you do the problem using only the notes.) Regardless,
you must give your solution with notation that is consistent with the notation we use
below.
(b) Consider the Cholesky Factorization Theorem discussed below (modified for real- valued
matrices):
Theorem. Given a SPD matrix A, there exists a lower triangular matrix L such
that A = LLT . If the diagonal elements of L are restricted to be positive, L is
unique.
We prove this theorem by showing that the right-looking Cholesky factorization
algorithm is well-defined for a matrix A that is SPD. Prove this theorem instead by
showing that the bordered Cholesky factorization algorithm is well-defined for a matrix
A that is SPD. In other words, show that every operation that is performed is legal.
Here are some hints:
(Although, you’ll learn more if you do the problem using only the notes.) Regardless, you
must given your solution with notation that is consistent with the notation we use below.