I'd like to expand a real, symmetric and positive definite Matrix $M$ into a Taylor series. I'm trying to do
$$ M (T) = M(T_0) + \frac{\partial M}{\partial T} (T-T_0) + \cdots$$
$T$ is a algebraic vector of parameters (e.g. temperatures at finite element nodes). I'm only interested in the first order term, i.e. the derivative of $M$ at $T_0$. My Professor tried to do this, but used unclear notation.
Best Answer
The derivative of $M(T)$ is simply the component-wise derivative. In the $2\times2$ case: $$\frac{d}{dT}\left(\matrix{a(T) & b(T) \cr c(T) & d(T)}\right) = \left(\matrix{a'(T) & b'(T) \cr c'(T) & d'(T)}\right)$$