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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.

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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.

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    $\begingroup$ I disagree on "When it comes to functions with matrices as an argument, they are defined only by the Taylor expansion of the function". What happens if the function does not admit Taylor expansion? There are other definitions. In QM Taylor expansion works for finite dimensional Hilbert spaces (spin spaces) and for analytic functions...The general case is handled via spectral theory. $\endgroup$ – Valter Moretti Jul 3 '17 at 6:44

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