Differential of a matrix function

Question

Let $n \geq 1$. Calculate the differential of the function G:
$$G: M_n( \mathbb{R}) \times M_n( \mathbb{R}) \rightarrow M_n( \mathbb{R})$$ $$(A,B)\rightarrow A^tBA$$

My first thought was to add small matrix variations $H$ and $K$ to $A$ and $B$ and then try to write as in the definition: $$D_{(A,B)}G=A^tBA + Lin(H,K)+o(||(H,K)||)$$
But then I end up with a long string of terms in $H,K$ and I can't determine which are the ones that fall into the Landau $o(||(H,K)||)$ and which are part of the linear expression.

So, could anyone explain to me where "to draw the line"?

Answer 1

The matrix $( D_{(A,B)} ) G((H,K)) $ is the directional derivative of $D$ at $(A,B)$ in the directions $(H,K)$. That is,

$$ ( D_{(A,B)} )G(H,K) = \frac{d}{ds} G(A + sH, B + sK)|_{s = 0}. $$

Write $G(A + sH, B + sK)$ explicitly, ignoring terms of order $s^2$ or $s^3$ as they will drop when differentiating and then differentiate and plug in $s = 0$.

Answer 2

I would proceed as you have, just whenever you have $H^2$, $HK$ or $K^2$ ditch it on sight. In fact, you don't even need to write them out when you expand: start the expansion, and just don't write the second order terms. Should drop out fairly easily.

Answer 3

The differential of a product is found by varying the factors one-at-a-time, while holding the other factors constant. $$\eqalign{ G &= A^TB\,A \cr dG &= dA^TB\,A + A^TdB\,A + A^TB\,dA \cr }$$ This expression is quite general. It holds for scalars, vectors, and matrices and under a variety of product types, e.g. Kronecker, Hadamard, Frobenius, dot, cross.

In the case of non-commutative products, you must also respect the ordering of the factors.