Block reflector

Template:More citations needed "A block reflector is an orthogonal, symmetric matrix that reverses a subspace whose dimension may be greater than one."<ref name="Backus.1969"> Template:Cite journal</ref>

It is built out of many elementary reflectors.

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 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.
• "*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

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