Page 483 - Applied Numerical Methods Using MATLAB

472 DIFFERENTIATION WITH RESPECT TO A VECTOR
Especially for a square, symmetric matrix A with M = N,we have

∂ T T if A is symmetric
x Ax = (A + A )x −−−−−−−−−→ 2Ax (C.6)
∂x

The second derivative of a scalar function f(x) with respect to a vector x =
T
[x 1 x 2 ] is called the Hessian of f(x) and is defined as
2
d 2 ∂ f/∂x 2 2
2 1 ∂ f/∂x 1 ∂x 2
H(x) =∇ f(x) = f(x) = 2 2 2 (C.7)
dx 2 ∂ f/∂x 2 ∂x 1 ∂ f/∂x
2
Based on this definition, we can write the following equation:
d 2 T T if A is symmetric
x Ax = A + A −−−−−−−−−→ 2A (C.8)
dx 2
On the other hand, the first derivative of a vector-valued function f(x) with
T
respect to a vector x = [x 1 x 2 ] is called the Jacobian of f(x) and is defined as
d ∂f 1 /∂x 1 ∂f 1 /∂x 2
J(x) = f(x) = (C.9)
dx ∂f 2 /∂x 1 ∂f 2 /∂x 2