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Cholesky Decomposition

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Please provide detailed proof.

7. 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.

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This solution shows step-by-step workings to contextualize the Cholesky Decomposition.

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