Page 482 - Applied Numerical Methods Using MATLAB

APPENDIX C

DIFFERENTIATION WITH

RESPECT TO A VECTOR

The first derivative of a scalar-valued function f(x) with respect to a vector
T
x = [x 1 x 2 ] is called the gradient of f(x) and defined as

d ∂f/∂x 1
∇f(x) = f(x) = (C.1)
dx ∂f/∂x 2
Based on this definition, we can write the following equation.

∂ T ∂ T ∂ y 1
x y = y x = (x 1 y 1 + x 2 y 2 ) = = y (C.2)
∂x ∂x ∂x y 2
∂ T ∂ 2 2 x 1
x x = (x + x ) = 2 = 2x (C.3)
1
2
∂x ∂x x 2
Also with an M × N matrix A,we have
∂ T ∂ T T
x Ay = y A x = Ay (C.4a)
∂x ∂x
∂ T ∂ T T T
y Ax = x A y = A y (C.4b)
∂x ∂x
where
M N

T
x Ay = a mn x m y n (C.5)
m=1 n=1

Applied Numerical Methods Using MATLAB , by Yang, Cao, Chung, and Morris
Copyright  2005 John Wiley & Sons, I nc., ISBN 0-471-69833-4

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