Editing Householder transformation (section)

== Computational and theoretical relationship to other unitary transformations ==
{{see also|Rotation (mathematics)}}
The Householder transformation is a reflection about a hyperplane with unit normal vector <math display="inline">v</math>, as stated earlier. An <math display="inline">N</math>-by-<math display="inline">N</math> [[unitary transformation]] <math display="inline">U</math> satisfies <math display="inline">UU^* = I</math>. Taking the determinant (<math display="inline">N</math>-th power of the geometric mean) and trace (proportional to arithmetic mean) of a unitary matrix  reveals that its eigenvalues <math display="inline">\lambda_i</math> have unit modulus. This can be seen directly and swiftly:
:<math>\begin{align}
  \frac{\operatorname{Trace}\left(UU^*\right)}{N} &=
  \frac{\sum_{j=1}^N\left|\lambda_j\right|^2}{N} = 1, &
  \operatorname{det}\left(UU^*\right) &=
  \prod_{j=1}^N \left|\lambda_j\right|^2 = 1.
\end{align}</math>

Since arithmetic and geometric means are equal if the variables are constant (see [[inequality of arithmetic and geometric means]]), we establish the claim of unit modulus.

For the case of real valued unitary matrices we obtain [[orthogonal matrices]], <math display="inline">UU^\textsf{T} = I</math>. It follows rather readily (see [[Orthogonal matrix]]) that any orthogonal matrix can be [[QR decomposition#Using Givens rotations|decomposed]] into a product of 2-by-2 rotations, called [[Givens rotation]]s, and Householder reflections. This is appealing intuitively since multiplication of a vector by an orthogonal matrix preserves the length of that vector, and rotations and reflections exhaust the set of (real valued) geometric operations that render invariant a vector's length.

The Householder transformation was shown to have a one-to-one relationship with the canonical coset decomposition of unitary matrices defined in group theory, which can be used to parametrize unitary operators in a very efficient manner.<ref>{{cite journal
 |author1=Renan Cabrera |author2=Traci Strohecker |author3=[[Herschel Rabitz]] |title= The canonical coset decomposition of unitary matrices through Householder transformations
 |journal=[[Journal of Mathematical Physics]]
 |volume=51 |issue=8 |pages=082101 |year=2010 
 |doi=10.1063/1.3466798 
 |arxiv=1008.2477|bibcode=2010JMP....51h2101C
|s2cid=119641896 }}</ref>

Finally we note that a single Householder transform, unlike a solitary Givens transform, can act on all columns of a matrix, and as such exhibits the lowest computational cost for QR decomposition and tridiagonalization. The penalty for this "computational optimality" is, of course, that Householder operations cannot be as deeply or efficiently parallelized. As such Householder is preferred for dense matrices on sequential machines, whilst Givens is preferred on sparse matrices, and/or parallel machines.