v2009.01.01 - Convex Optimization

384 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX
5.5.2 Rotation/Reflection
Rotation of the list X ∈ R n×N about some arbitrary point α∈ R n , or
reflection through some affine subset containing α , can be accomplished
via Q(X −α1 T ) where Q is an orthogonal matrix (B.5).
We rightfully expect
D ( Q(X − α1 T ) ) = D(QX − β1 T ) = D(QX) = D(X) (883)
Because list-form D(X) is translation invariant, we may safely ignore
offset and consider only the impact of matrices that premultiply X .
Interpoint distances are unaffected by rotation or reflection; we say,
EDM D is rotation/reflection invariant. Proof follows from the fact,
Q T =Q −1 ⇒ X T Q T QX =X T X . So (883) follows directly from (798).
The class of premultiplying matrices for which interpoint distances are
unaffected is a little more broad than orthogonal matrices. Looking at EDM
definition (798), it appears that any matrix Q p such that
X T Q T pQ p X = X T X (884)
will have the property
D(Q p X) = D(X) (885)
An example is skinny Q p ∈ R m×n (m>n) having orthonormal columns. We
call such a matrix orthonormal.
5.5.2.0.1 Example. Reflection prevention and quadrature rotation.
Consider the EDM graph in Figure 104(b) representing known distance
between vertices (Figure 104(a)) of a tilted-square diamond in R 2 .
Suppose some geometrical optimization problem were posed where isometric
transformation is feasible excepting reflection, and where rotation must be
quantized so that only quadrature rotations are feasible; only multiples
of π/2.
In two dimensions, a counterclockwise rotation of any vector about the
origin by angle θ is prescribed by the orthogonal matrix
[ ]
cos θ − sin θ
Q =
sin θ cos θ
(886)