TITLE: A Householder matrix is symmetric
QUESTION [5 upvotes]: I want to show that a Householder matrix is symmetric, so I must show that $H^T = H$, but from the formula
$$H= I - (uu^T/\beta),$$
they are not equal. What's wrong with my reasoning?
EDIT: I forgot that $(uu^T)^T$ would be $(u^T)^T(u)^T$ from the following properties:
 $(AB)^T=B^TA^T$

REPLY [2 votes]: $$H = I - \frac {2} {u^T u} u u^T=I+\alpha \ u u^T$$
$$H^T=(I+\alpha \ u u^T)^T=I^T+\alpha \ (uu^T)^T=I+\alpha (u^T)^Tu^T=I+\alpha \ u u^T=H$$
Where I used $I^T=I$ and the basic properties of the transposed matrix, namely:

For scalars $\lambda$ we have $(\lambda A)^T=\lambda A^T$
$(A+B)^T=A^T+B^T$
$(AB)^T=B^TA^T$
$(A^T)^T=A$