Taylor expansion of a function whose arguments are matrices

Question

I'm studying rotations expressed by a matrix $R$.

$|\psi> \to \hat{U} (R) |\psi>$

When we assume infinitesimal rotations, we can write $R = E + \omega$ where $E$ is an identity matrix and $\omega$ is a real matrix. Then, according to "Lectures on Quantum Mechanics" by S.Weinberg, $\hat{U} (E + \omega)$ must take the form

$\hat{U} (E + \omega) = E + \frac{i}{2\hbar} \sum _{ij} \omega _{ij} J_{ij} + O(\omega ^2).$

But why? My teacher said this is taylor expansion but I've never heard of taylor expansion of matrix functions.

Answer 1

When it comes to functions with matrices as an argument, they are defined only by the Taylor expansion of the function.

In general, functions of a real or (especially) complex number can be considered as defined by their Taylor series, and so if one wants to consider functions of a matrix, then the Taylor expansion is the way to go.

This is because all we can do with matrices is multiply, add, and invert them (if they can be inverted), so we need to make functions that only use these operations, and the Taylor series only involve adding and multiplying the arguments.