The Cholesky decomposition of a positive definite matrix is the unique lower triangular matrix L =

The Cholesky decomposition of a positive definite matrix Ω is the unique lower triangular matrix L = (lij ) satisfying Ω = LLt and lii > 0 for all i. Let Φ be the kinship matrix of a pedigree with n people numbered so that parents precede their children. The Cholesky decomposition L of Φ can be defined inductively one row at a time starting with row 1.

Given that rows 1,...,i − 1 have been defined and that i has parents r and s, define [4, 10]

lij =







0 j>i 1

2 lrj + 1 2 lsj j

(Φii − i−1 k=1 l 2

ik)

1 2 j = i.

Prove by induction that L is the Cholesky decomposition of Φ. Why is lii positive? (Hints: Φii > 1 2Φri + 1 2Φsi and Φij = 1 2Φrj + 1 2Φsj for j