Cholesky decomposition

Question

Question:

Cholesky Decomposition

Please provide detailed proof.

1. Let k and l be positive integers, and set n:= k + l. Suppose A ∈ C^nxn has the decomposition

                         A = | B    C^H |
                               | C      D |,

where B ∈C^kxk, B ∈C^lxk, and D∈C^lxl.

(a) If A is HPD, prove that B, D and E :: D - (C B^-1 C^H) are HPD. E is called the Schur compliment of B in A.

(b) Suppose A is HPD. Express the Cholesky factorization of A in terms of the Cholesky factorizations of B, D, and E.