Block reflector

"A block reflector is an orthogonal, symmetric matrix that reverses a subspace whose dimension may be greater than one."¹

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

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.