householder Householder orthogonal reflexion matrix Syntax householder() // demo u = householder(v [,w]) [u, H] = householder(v [,w]) Arguments v real or complex column vector w 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. Description householder(..) 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 History 6.0 Householder reflexion matrix added as second output parameter. Demo householder() added. Help page reviewed.