Derivative of Matrix Inverse with Respect to Scalar

Question

Let $X$ be a positive-definite matrix. Let $\alpha$ be a scalar variable. What is $\frac{\partial f}{\partial \alpha}$ when $f = (I + \alpha X)^{-1}$?

What if $X$ is any matrix (not restricted to the set of positive-definite matrices)?

Answer 1

Note that $(I+\alpha X) f(\alpha) = I$, so $X f(\alpha) + (I+\alpha X) f'(\alpha) = 0$, hence $f'(\alpha) = - f(\alpha) X f(\alpha)$.

The same formula will hold as long as the inverse of $I+ \alpha X$ exists (since it is invertible locally).