Block reflector

"A block reflector is an orthogonal, symmetric matrix that reverses a subspace whose dimension may be greater than one."

• see abstract here http://locus.siam.org/SINUM/volume-25/art_0725014.html

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 $Q$ belonging to $\mathcal M_n(\R)$ can be written in the form : $Q = I -auu^T$ where $I$ is the identity matrix for $\mathcal M_n(\R)$ , $a$ is a scalar and $u$ belongs to $\mathcal \R^n$ .

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