Let f: R^n -> R be defined as follows:
f(x) = x*L(x) where * denotes the standard inner product and L: R^n -> R^n is a linear
I'm trying to find the directional derivative f'(x;u).
I know that f'(x;u) (the directional derivative of f(x) in the direction of the unit vector u)
is equal to gradient(f(x)) * u where * denotes inner product.
The Attempt at a Solution
In this case gradient(f(x)) = gradient(x*L(x)) = gradient(x)*L(x) + L(x)*gradient(x)
(Not sure if this step is correct).
Then because L: R^n -> R^n is a linear map the differential of L is L again, no?
So we get: gradient(f(x)) = gradient(x)*L(x) + L(x)*gradient(x) = 2L(x)*gradient(x).
Conclusion: the directional derivative in the direction of u is then (2L(x)*gradient(x))*u where u is a unit vector.
Is this correct? I'm a little bit confused about the part of L being a linear map, is it correct
to state that the gradient is again L itself?