matrix norm derivative with respect a parameter

Question

What is the result of the following expression $\frac{d}{dt}\left( \|A(t)-B(t) \|\right) $, where $\|\cdot \|$ can be for instance the Frobenius norm?

Answer 1

Let $\phi(t) = \langle X(t),X(t) \rangle $, where $X(t) = A(t)-B(t)$ and $\langle C,D \rangle = \operatorname{tr}(C^T D)$.

Then we have $\phi'(t) = 2 \langle X'(t),X(t) \rangle = 2\langle A'(t)-B'(t),A(t)-B(t) \rangle $.

Since $\eta(t)= \|A(t)-B(t)\| = \sqrt{\phi(t)}$, we have $\eta'(t) = {\operatorname{tr} ((A'(t)-B'(t))^T(A(t)-B(t)) ) \over \|A(t)-B(t)\|} $ (assuming $A(t) \neq B(t)$).