3D Orthogonal Matrix
If A ? C nxn, B ? C pxp, and X ? C nxp, satisfy AX = XB, rank(X) = p then there exists a unitary Q ? C nxn such that QHAQ =
- (Golub & Van Loan, page 334)
Q is the ‘Q’ in the QR decomposition of X
In our case, B = ? and p = 1 with A x = ? x
- Q is the P in Px = ||x||2 [1, 0, 0]
- P is the Householder matrix I - 2 v vT / vT v
- Where v = x + sign(x1) ||x||2 [1, 0, 0]