:Block reflector

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

{{Cite journal

| last1 = Schreiber

| first1 = Rober

| last2 = Parlett

| first2 = Beresford

| year = 2006

| url = http://epubs.siam.org/doi/abs/10.1137/0725014

| title = Block Reflectors: Theory and Computation

| journal = SIAM Journal on Numerical Analysis

| volume = 25

| pages = 189–205

| doi = 10.1137/0725014

}}

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 $\mathcal M_n(\R)$ can be written in the form :

$Q = I -auu^T$ where $I$ is the identity matrix for $\mathcal M_n(\R)$ , $a$ is a scalar and $u$ belongs to $\R^n$ .

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.