Linear algebra – Hermitian property of a householder transformation on a complex field

To let $ mathbf {H} $ a householder (i.e., elementary reflector) so that $ mathbf {Hx} = mathbf {e} _1 $, for a $ mathbf {x} in Bbb {C} ^ n $,

I have defined that $ mathbf {H} = e ^ {i theta} left ( mathbf {I} – 2 frac { mathbf {u} mathbf {u} ^ *} { mathbf {u} ^ * mathbf {u}} right) $, from where $ e ^ {i theta} = frac { bar {x_1}} {| x_1 |} $ and $ mathbf {u} = mathbf {x} – e ^ {i theta} | mathbf {x} | _2 mathbf {e} _1 $, I actually get the expected result from $ mathbf {Hx} = mathbf {e} _1 $but my trouble is finding out how to enforce the property $ mathbf {H} = mathbf {H} ^ * $, Can someone please show me the right direction?

I checked that too $ mathbf {H} $ is uniform, there $ mathbf {H} ^ * = mathbf {H ^ {- 1}} = e ^ {- i theta} left ( mathbf {I} – 2 frac { mathbf {u} mathbf {u } ^ *} { mathbf {u} ^ * mathbf {u}} right) $ and $ mathbf {H} ^ * mathbf {H} = mathbf {I} $but I do not see how $ mathbf {H} = mathbf {H ^ {- 1}} $, I even understand that $ mathbf {H} ^ {- 1} mathbf {e} _1 = mathbf {x} $what should be expected.

Any help is greatly appreciated.