Editing Householder transformation (section)

====Properties====

The Householder matrix has the following properties:
* it is [[Hermitian matrix|Hermitian]]: <math display="inline">P = P^*</math>,
* it is [[unitary matrix|unitary]]: <math display="inline">P^{-1} = P^*</math> (via the [[Sherman-Morrison formula]]),
* hence it is [[involutory matrix|involutory]]: <math display="inline">P = P^{-1}</math>.
* A Householder matrix has eigenvalues <math display="inline">\pm 1</math>.  To see this, notice that if <math display="inline">\vec x</math> is orthogonal to the vector <math display="inline">\vec v</math> which was used to create the reflector, then <math display="inline">P_v\vec x = (I-2\vec v\vec v^*)\vec x = \vec x-2\langle\vec v,\vec x\rangle\vec v = \vec x</math>, i.e., <math display="inline">1</math> is an eigenvalue of multiplicity <math display="inline">n - 1</math>, since there are <math display="inline">n - 1</math> independent vectors orthogonal to <math display="inline">\vec v</math>.  Also, notice <math display="inline">P_v\vec v = (I-2\vec v\vec v^*)\vec v = \vec v - 2\langle\vec v,\vec v\rangle\vec v = -\vec v</math> (since <math>\vec v</math> is by definition a unit vector), and so <math display="inline">-1</math> is an eigenvalue with multiplicity <math display="inline">1</math>.
* The [[determinant]] of a Householder reflector is <math display="inline">-1</math>, since the determinant of a matrix is the product of its eigenvalues, in this case one of which is <math display="inline">-1</math> with the remainder being <math display="inline">1</math> (as in the previous point), or via the [[Matrix determinant lemma]].