I'm reading a book on quantum computing. In the book it says that any linear optical element (represented by the set of unitary matrices $U(2)$) is equivalent to a combination of balanced phase shifters and beamsplitters. The matrix for a balanced phase shifter is
$B_{l}=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \\ \end{pmatrix}$
The matrix for a phase shifter is
$P=\begin{pmatrix} e^{i\phi} & 0 \\ 0 & 1 \\ \end{pmatrix}$ or $P=\begin{pmatrix} 1 & 0 \\ 0 & e^{i\theta} \\ \end{pmatrix}$
Is there any way I can prove this mathematically? i.e. $\forall A\in U(2)$, $A$ can be written as products of $B_{l}$ and $P$
Edit: Here's what I did before posting this question. I know that $U(2)$ is isomorphic to the semi-direct product of $U(1)$ and $SU(2)$. So I tried to construct those two groups using $B_{l}$ and $P$, but I feel like I'm overthinking.