Jelly, 13 bytes

Ø.,æ*Æṭ,ḢN*Nʋ

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Takes input as [v, u] on the left, and a on the right. Outputs as [α, β]

Uses xnor's formula that

$$\alpha = \operatorname{tr} \left( \begin{bmatrix} 0 & 1 \\ v & u \end{bmatrix}^a \right) \\ \beta = -(-v)^a$$

Full explanation coming