Block reflector

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

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

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