Editing Block reflector

{{More citations needed|date=May 2021}}
"A '''block reflector''' is an orthogonal, symmetric [[matrix (mathematics)|matrix]] that reverses a subspace whose dimension may be greater than one."<ref name="Backus.1969">
{{Cite journal
| last1 = Schreiber
| first1 = Rober
| last2 = Parlett
| first2 = Beresford
| year = 2006
| url = http://epubs.siam.org/doi/abs/10.1137/0725014
| title = Block Reflectors: Theory and Computation
| journal = SIAM Journal on Numerical Analysis
| volume = 25
| pages = 189–205
| doi = 10.1137/0725014
}}</ref>

It is built out of many [[elementary reflector]]s.

It is also referred to as a [[triangular factor]], and is a [[triangular matrix]] and they are used in the [[Householder transformation]].

A reflector <math> Q </math> belonging to <math>\mathcal M_n(\R) </math> can be written in the form :
<math> Q = I -auu^T  </math> where <math>I</math> is the [[identity matrix]] for <math>\mathcal M_n(\R) </math>, <math>a</math> is a [[Scalar (mathematics)|scalar]] and <math>u</math> belongs to <math>\R^n</math> .

==LAPACK routines==
Here are some of the [[LAPACK]] routines that apply to block reflectors
* "*larft" forms the triangular vector T of a block reflector H=I-VTVH.
* "*larzb" applies a block reflector or its [[transpose]]/[[conjugate transpose]] as returned by "*tzrzf" to a general [[matrix (mathematics)|matrix]].
* "*larzt" forms the triangular vector T of a block reflector H=I-VTVH as returned by "*tzrzf".
* "*larfb" applies a block reflector or its [[transpose]]/[[conjugate transpose]] to a general rectangular matrix.

==See also==
* [[Reflection (mathematics)]]
* [[Householder transformation]]
* [[Unitary matrix]]
* [[Triangular matrix]]

==References==
{{Reflist}}

[[Category:Matrices (mathematics)]]


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