Prove that any pair of nonnegative numbers (2 a, 2 d) can be realized as additive and

Prove that any pair of nonnegative numbers (σ2

a, σ2

d) can be realized as additive and dominance genetic variances. The special pairs ( 1 2 , 0)

and (0, 1) show that the two matrices Φ = (Φij ) and ∆7 = (∆7ij )

defined for an arbitrary non-inbred pedigree are legitimate covariance matrices. (Hint: Based on the previous problem,

σ2 a = 2p1p2(p1u + p2v)

2

σ2 d = p2 1p2 2(u − v)

2 for u = µ11 − µ12 and v = µ12 − µ22 Solve for u and v.)