I know any unitary matrix can be factored as follows: $$\underline {\overline {\bf{U}} } = \left( {\prod\limits_{j = N}^1 {\underline {\overline {\bf{\Psi }} } \left( {{{\underline {\bf{w}} }_j}} \right)} } \right)\underline {\overline {\bf{\Phi }} }$$ Here, each $$\underline {\overline {\bf{\Psi }} } \left( {{{\underline {\bf{w}} }_i}} \right) = \left[ {\begin{array}{*{20}{c}} {{{\underline {\overline {\bf{I}} } }_{\left[ {i - 1} \right]}} - \frac{{{{\underline {\bf{w}} }_i}\underline {\bf{w}} _i^ + }}{{1 + \sqrt {1 - \underline {\bf{w}} _i^ + {{\underline {\bf{w}} }_i}} }}}&{{{\underline {\bf{w}} }_i}}&{\underline {\overline {\bf{0}} } }\\ { - \underline {\bf{w}} _i^ + }&{\sqrt {1 - \underline {\bf{w}} _i^ + {{\underline {\bf{w}} }_i}} }&{{{\underline {\bf{0}} }^ + }}\\ {\underline {\overline {\bf{0}} } }&{\underline {\bf{0}} }&{{{\underline {\overline {\bf{I}} } }_{\left[ {N - i} \right]}}} \end{array}} \right]$$ is a primitive unitary matrix depending only upon the $i - 1{\rm{ }}x{\rm{ }}1$ complex vector $${\underline {\bf{w}} _i} = \left[ {\begin{array}{*{20}{c}} {{w_{1i}}}\\\vdots \\{{w_{\left( {i - 1} \right)i}}} \end{array}} \right]$$ with the only constraint on ${\underline {\bf{w}} _i}$ being that $\left| {{{\underline {\bf{w}} }_i}} \right| \le 1$; $\underline {\overline {\bf{\Phi }} }$ is an $N{\rm{ }}x{\rm{ }}N$ diagonal phase matrix, i.e. $$\underline {\overline {\bf{\Phi }} } = \left[ {\begin{array}{*{20}{c}} {{e^{j{\varphi _1}}}}& \cdots &0\\ \vdots & \ddots & \vdots \\ 0& \cdots &{{e^{j{\varphi _N}}}} \end{array}} \right]$$ See here for a proof of this assertion.

The real case is analogous, other than each ${\underline {\bf{w}} _i}$ is real rather than complex, and $\underline {\overline {\bf{\Phi }} }$ reduces to a sign matrix (i.e., all $\pm 1$ entries).

Does this factorization have a standard name?


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