Page 158 - Applied Probability

8. The Polygenic Model
143
(a) If B =(b ij ) is a square matrix with cofactor B ij corresponding to
entry b ij , then the determinant det B =
b ij B ij is expandable on

j
−1
any row i.If B is invertible as well, then its inverse C = B
has
1
(B ji ).
entries c ij =
det B
(b) If B =(b ij ) is a square matrix, then the trace tr(B)of B is defined

by tr(B)= i ii . The trace function satisfies tr(BC) = tr(CB) for
b
any two conforming matrices B and C.
t
t
t
(c) The matrix transpose operation satisfies (BC) = C B .
t
(d) The expectation of a random vector X =(X 1 ,...,X n ) is defined com-
t
ponentwise by E(X)= [E(X 1 ),..., E(X n )] . Linearity carries over
from the scalar case in the sense that
E(X + Y ) = E(X)+E(Y )
E(BX)= B E(X)
for a compatible random vector Y and a compatible matrix B.
(e) If B is a matrix and W is a random vector, then the quadratic form
t
t
t
W BW has expectation E(W BW) = tr[B Var(W)]+E(W) B E(W).
To verify this assertion, observe that

t
E(W BW) = E W i b ij W j
ij

= b ij E(W i W j )
ij

= b ij [Cov(W i ,W j )+ E(W i )E(W j )]
ij
t
= tr[B Var(W)] + E(W) B E(W).
(f) The partial derivative of a matrix B =(b ij ) with respect to a scalar
parameter θ is the matrix with entries ( ∂ b ij ). Because the trace func-
∂θ
tion is linear, ∂ tr(B) = tr( ∂ B). The product rule of differentiation
∂θ ∂θ
implies ∂ (BC)= ( ∂ B)C + B ∂ C.
∂θ ∂θ ∂θ
(g) The derivative of a matrix inverse is ∂ B −1 = −B −1 ( ∂ B)B −1 .To
∂θ ∂θ
derive this formula, solve for ∂ B −1 in
∂θ
∂
0= I
∂θ
∂ −1
= (B B)
∂θ
∂ −1 −1 ∂

= B B + B B.
∂θ ∂θ