Some Properties and Applications of Simple Orthogonal Matrices

Conditions are found for a general transformation in the planeof two vectors u and v to be orthogonal. The results characterizea rotation in the (u, v)-plane by the angle ø betweenu and v and the angle of rotation. When ø = /2 the Jacobirotation matrix is a special case, but other choices of øare interesting. The rotation that maps a single vector x intoa vector y of the same size, by rotating in the (x, y)-plane,is found and this may be used in much the same way that Householdertransforms are used. If (x₁, y₁) and (x₂, y₂) are pairs of vectorscompatible in size and angle, the orthogonal matrix that rotatesin a suitably chosen plane so that x₁ x₂ and y₁ y₂ is found.This has applications in mapping two columns of a matrix toa simple form, similar to Householder operations on a singlecolumn.