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

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