Computing derivative with vectors and matrices

Question

I do not have much knowledge on derivatives for vectors and matrices. Could someone please help me computing the derivative of following equation and showing me all the steps, please?

$f(\theta) = \langle a, A \theta \rangle + \lambda ( \langle (\theta^+ - \theta), A(\theta^+ - \theta) \rangle - \alpha )$

where $a, \theta, \theta^*$ are vectors of size $d$, A a matrix of size $d \times d$, and $\lambda, \alpha$ are scalars.

Answer 1

Here are a couple of hints. The derivative is still linear, so l writing:

$f = f_1 + f_2$

For the two expressions. We have

$df = df_1 + df_2$

So we can work out the derivative of each function seperately. Second, we have:

$ d<f,g> = <df, g> + <f, dg>$

And also for $A$ a constant matrix and so representing a linear map:

$dA = A$

Then:

$d<a, A\theta>$

$= <da, A\theta> + <a, d(A\theta)>$

$ = 0 + <a, A\theta>$

$= <a,A\theta>$