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 ${\displaystyle \mathbb {R} ^{n}}$ .

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

1. Schreiber, Rober; Parlett, Beresford (2006). "Block Reflectors: Theory and Computation" . SIAM Journal on Numerical Analysis. 25: 189–205. doi:10.1137/0725014.