2.2 Derivatives of an Inverse 2 DERIVATIVES
2.1.3 Square forms
If X is square and invertible, then
∂ det(X
T
AX)
∂X
= 2 det(X
T
AX)X
−T
If X is not square but A is symmetric, then
∂ det(X
T
AX)
∂X
= 2 det(X
T
AX)AX(X
T
AX)
−1
If X is not square and A is not symmetric, then
∂ det(X
T
AX)
∂X
= det(X
T
AX)(AX(X
T
AX)
−1
+ A
T
X(X
T
A
T
X)
−1
) (13)
2.1.4 Other nonlinear forms
Some special cases are (See [8])
∂ ln det(X
T
X)|
∂X
= 2(X
+
)
T
∂ ln det(X
T
X)
∂X
+
= −2X
T
∂ ln |det(X)|
∂X
= (X
−1
)
T
= (X
T
)
−1
∂ det(X
k
)
∂X
= k det(X
k
)X
−T
See [7].
2.2 Derivatives of an Inverse
From [15] we have the basic identity
∂Y
−1
∂x
= −Y
−1
∂Y
∂x
Y
−1
from which it follows
∂(X
−1
)
kl
∂X
ij
= −(X
−1
)
ki
(X
−1
)
jl
∂a
T
X
−1
b
∂X
= −X
−T
ab
T
X
−T
∂ det(X
−1
)
∂X
= −det(X
−1
)(X
−1
)
T
∂Tr(AX
−1
B)
∂X
= −(X
−1
BAX
−1
)
T
Petersen & Pedersen, The Matrix Cookbook (Version: January 5, 2005), Page 8
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