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I was studying the Glashow-Weinberg-Salam theory and I have found this relation:

$$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} = e^{\frac{i\beta}{2}}\,e^{-\frac{i\alpha_3}{2}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix}$$

I do not know how the exponential matrix acts on the vector.

Thanks in advance!