Find a unitary matrix $U$ such that $Ux = x$ and $Uy = -y$ for given unit perpendicular vectors $x$ and $y$ of $\mathbb{R}^{n}$

Question

Find a unitary matrix $\bf{U}$ such that $\bf{U}$$x = x$ and $\bf{U}$$y = -y$ for given $\bf {unit}$ perpendicular (orthogonal) vectors x and y of $R^{n}$

If matrix is not found then please tell me why such a unitary matrix exists.

Answer 1

Since $x$ and $y$ are unit vectors and perpendicular, let $U=[x|y|S],$ where $S\in\mathbb{R}^{n\times(n-2)}$ has columns which are an orthonormal basis for the orthogonal complement of $\mathrm{span}(\{x,y\}).$ Then $U$ is unitary, and clearly $U^{*}x=e_{1},$ $U^{*}y=e_{2},$ where $e_{j}$ denotes the standard $j$th unit vector. We observe that the matrix $A=\mathrm{diag}(1,-1,1,\ldots,1)$ is unitary, and $Ae_{1}=e_{1},$ $Ae_{2}=-e_{2}.$ Then $$UAU^{*}x=UAe_{1}=Ue_{1}=x,$$ and $$UAU^{*}y=UAe_{2}=U(-e_{2})=-Ue_{2}=-y,$$ and since the product of unitary matrices is unitary, $UAU^{*}$ has all of the desired properties.