Block reflector explained

"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

belonging to can be written in the form : where is the identity matrix for , is a scalar and belongs to .

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

Notes and References

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