householderHouseholder orthogonal reflexion matrixSyntax
householder() // demo
u = householder(v [,w])
[u, H] = householder(v [,w])
Argumentsvreal or complex column vectorw
real or complex column vector with same size as v.
Default value is eye(v) ((Ox) axis).
u
unit vector lying in the (v,w) plane and orthogonal
to the bisectrix of (v,w).
Column of size(v) of real or complex numbers.
H
Orthogonal Householder reflexion matrix: H= eye() - 2*u*u'.
H is such that inv(H)==H,
H'==H, and det(H)==-1.
If v and w are real,
H*v is proportional to w.
Descriptionhouseholder(..) computes the unit vector u
lying in the (v,w) plane and orthogonal to the bisectrix of
(v,w).
If v and w are proportional:
If they are opposite, u= v/|v| is returned.
If they are real and have the same direction, u is set
in the (xOy) plane with a priori u(1)>0, and orthogonal to
v (u'*v==0). However,
If they are along (Ox), u = (Oy+) is returned instead.
If v and w are scalars with same
signs, the orthogonal sub-space is restricted to {0}
that can't be normalized:
u and H are then set to
%nan.
If the related reflexion matrix H is computed, for any point A
of column coordinates a, H*a are the coordinates of
the reflected image of A (see the example below).
If v or/and w are in row, they are priorly
transposed into columns.
If v or/and w are [],
[] is returned for u and H.
Examples Application: Reflected image of an object w.r. to a given plane
See Also
qr
givens
History6.0Householder reflexion matrix added as second output parameter.
Demo householder() added. Help page reviewed.