If we consider two complex Higgs doublets with $SU(3)_{c} ⊗ SU(2)_L ⊗ U(1)_Y$ quantum numbers $\phi_{A,B} \sim (1,2,1)$, such that
$$\begin{align}&\phi_A = \begin{pmatrix} \phi_{A}^{\dagger}\\ \dfrac{1}{\sqrt2} (\operatorname{Re} \phi_{A}^{0} + i\operatorname{Im} \phi_{A}^{0}) \end{pmatrix} &\phi_A = \begin{pmatrix} \phi_{B}^{\dagger}\\ \dfrac{1}{\sqrt2} (\operatorname{Re} \phi_{B}^{0} + i\operatorname{Im} \phi_{B}^{0}) \end{pmatrix}\end{align}$$
If we assume that they acquire the vacuum expectation values (vevs):
$$\begin{align}&\langle\phi_A\rangle = \begin{pmatrix} 0\\ \dfrac{v_A}{\sqrt2} \end{pmatrix} &\langle\phi_B\rangle = \begin{pmatrix} 0\\ \dfrac{v_B}{\sqrt2} \end{pmatrix}\end{align}$$
I know that we can change the basis from $\{\phi_A,\phi_B\}$ to a $\{\phi_1,\phi_2\}$ basis, bearing in mind that only one doublet gains a non zero vev as follows:
$$\begin{align}&\langle\phi_1\rangle = \begin{pmatrix} 0\\ \dfrac{v}{\sqrt2} \end{pmatrix} &\langle\phi_2\rangle = \begin{pmatrix} 0\\ 0 \end{pmatrix}\end{align}$$
My issue is that which particular transformation can I use to be able to do this transformation and how do I proceed from there. Basically asking for how can I express $\{\phi_A,\phi_B\}$ in terms of $\{\phi_1,\phi_2\}$ and relate the $v_A, v_B $ and $v$ altogether
I would appreciate greatly if someone could help me out.
Thanks