Derivative of composition of real function and matrix calculation

Question

I have

$f: R^n \Rightarrow R^1$

and $H$ a $n \times n$ matrix (that is inversible)

What's the gradient of:

$g: x \mapsto f(H.x)$

expressed using the gradient of f? Is it just $x \mapsto H. \nabla f(H.x) $ ?

Answer 1

You get $\nabla g(x) = H^T\cdot \nabla f(H\cdot x)$ from $\nabla g(x) = (D g(x))^T = (D f(Hx)\cdot H)^T = H^T \cdot (D f(Hx))^T = H^T \cdot \nabla f(Hx)$.