## 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.