Householder transformation

In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder.[1]

Its analogue over general inner product spaces is the Householder operator.

DefinitionEdit

TransformationEdit

The reflection hyperplane can be defined by a unit vector   (a vector with length  ) that is orthogonal to the hyperplane. The reflection of a point   about this hyperplane is the linear transformation:

 

where   is given as a column unit vector with Hermitian transpose  .

Householder matrixEdit

The matrix constructed from this transformation can be expressed in terms of an outer product as:

 

is known as the Householder matrix, where   is the identity matrix

PropertiesEdit

The Householder matrix has the following properties:

  • it is Hermitian:  ,
  • it is unitary:  ,
  • hence it is involutory:  .
  • A Householder matrix has eigenvalues  . To see this, notice that if   is orthogonal to the vector   which was used to create the reflector, then  , i.e.,   is an eigenvalue of multiplicity  , since there are   independent vectors orthogonal to  . Also, notice  , and so   is an eigenvalue with multiplicity  .
  • The determinant of a Householder reflector is  , since the determinant of a matrix is the product of its eigenvalues, in this case one of which is   with the remainder being   (as in the previous point).

ApplicationsEdit

Geometric opticsEdit

In geometric optics, specular reflection can be expressed in terms of the Householder matrix (see Specular reflection § Vector formulation).

Numerical linear algebraEdit

Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and is the first step of the QR algorithm. They are also widely used for transforming to a Hessenberg form. For symmetric or Hermitian matrices, the symmetry can be preserved, resulting in tridiagonalization.[2]

QR decompositionEdit

Householder reflections can be used to calculate QR decompositions by reflecting first one column of a matrix onto a multiple of a standard basis vector, calculating the transformation matrix, multiplying it with the original matrix and then recursing down the   minors of that product.

TridiagonalizationEdit

This procedure is presented in Numerical Analysis by Burden and Faires.[3] In the first step, to form the Householder matrix in each step we need to determine   and  , which are:

 

From   and  , construct vector  :

 

where  ,  , and

  for each  

Then compute:

 

Having found   and computed   the process is repeated for   as follows:

 

Continuing in this manner, the tridiagonal and symmetric matrix is formed.

ExamplesEdit

In this example, also from Burdern and Faires[3], the given matrix is transformed to the similar tridiagonal matrix A3 by using the Householder method.

 

Following those steps in the Householder method, we have:

The first Householder matrix:

 

Used   to form

 

As we can see, the final result is a tridiagonal symmetric matrix which is similar to the original one. The process is finished after two steps.

Computational and theoretical relationship to other unitary transformationsEdit

The Householder transformation is a reflection about a hyperplane with unit normal vector  , as stated earlier. An  -by-  unitary transformation   satisfies  . Taking the determinant ( -th power of the geometric mean) and trace (proportional to arithmetic mean) of a unitary matrix reveals that its eigenvalues   have unit modulus. This can be seen directly and swiftly:

 

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,  . It follows rather readily (see orthogonal matrix) that any orthogonal matrix can be decomposed into a product of 2 by 2 rotations, called Givens Rotations, 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.[4]

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.

ReferencesEdit

  1. ^ Householder, A. S. (1958). "Unitary Triangularization of a Nonsymmetric Matrix" (PDF). Journal of the ACM. 5 (4): 339–342. doi:10.1145/320941.320947. MR 0111128.
  2. ^ Schabauer, Hannes; Pacher, Christoph; Sunderland, Andrew G.; Gansterer, Wilfried N. (2010-05-01). "Toward a parallel solver for generalized complex symmetric eigenvalue problems". Procedia Computer Science. 1 (1): 437–445. doi:10.1016/j.procs.2010.04.047.
  3. ^ a b Burden, Richard; Faires, Douglas (10 December 2004). Numerical analysis (8th ed.). Thomson Brooks/Cole. ISBN 9780534392000.
  4. ^ Renan Cabrera; Traci Strohecker; Herschel Rabitz (2010). "The canonical coset decomposition of unitary matrices through Householder transformations". Journal of Mathematical Physics. 51 (8): 082101. arXiv:1008.2477. Bibcode:2010JMP....51h2101C. doi:10.1063/1.3466798.