
Re: Nx2N lapped orthogonal transform
Posted:
Aug 20, 2013 6:24 AM


Am 19.08.2013 06:34, schrieb James Dow Allen: > Let > ( A B 0 ) > ( 0 A B ) > ( B 0 A ) > be a 3Nx3N real matrix with A,B,0 each NxN and 0 an allzeros matrix. > > What is the necessary and sufficient condition for that matrix to be > orthogonal, i.e. that its transpose also be its inverse? > > This problem statement can be considered ambiguous. > *But if you derive a good parametric form for (A B) you will know it.* > > (I already "know the answer." I post from curiosity: Is this a VERY easy > problem, or just an easy problem.)
Since orthogonality of a nxn real matrix X, X^+.X = Id is easier expressed by "the columns as well as the rows of the matrix X form an ndimensional orthonormal frame: X_i.X_k =delta_ik", the condition is simply that the three 6d vectors
{a_i1,..a_i3, b_i1..b_i3}, i=1,2,3
form an orthonormal tripod in 6 dimensions.
Parametrization may be done in descriptive words by "choose at random 3 different vectors from a unit vector in 6d spherical coordinates {theta1..theta5,phi}, and partition that matrix in 2 x 3 x 3".

Roland Franzius

