Some properties and applications of simple orthogonal matrices

Conditions are found for a general transformation in the plane of two vectors u and v to be orthogonal. The results characterize a rotation in the (u, v)-plane by the angle ø between u and v and the angle of rotation. When ø = π/2 the Jacobi rotation matrix is a special case, but other choices of ø are interesting. The rotation that maps a single vector x into a 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 Householder transforms are used. If (x1, y1) and (x2, y2) are pairs of vectors compatible in size and angle, the orthogonal matrix that rotates in a suitably chosen plane so that x1 → x2 and y1 → y2 is found. This has applications in mapping two columns of a matrix to a simple form, similar to Householder operations on a single column. RESP-7948 and RESP-7741