To show that the function $f:M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A)=AA^t$ is differentiable and evaluate its derivative

Question

How to show that the function $f:M_n(\mathbb R) \to M_n(\mathbb R)$ given by $f(A)=AA^t$ is differentiable and how to find the total differential at a point $X$ i.e. how to find $D f_A(X)$ ?

Answer 1

It is differentiable because it is a polynomial function in each of the coordinates. To find the differential, we can compute $$ f(A+\epsilon X)-f(A)=(A+\epsilon X)(A^T+\epsilon X^T)-AA^T=\epsilon (X A^T+AX^T)+O(\epsilon^2). $$ Therefore $Df_A(X)=XA^T+AX^T$.