It uses the (U, D) representation of generalized ellipsoids in \(R^d\),
where
\(U\) is square orthogonal and \(D\) is diagonal with extended non-negative real
numbers, i.e. 0, Inf or a positive real). These are roughly analogous to the
corresponding terms in the singular-value decomposition of a matrix,
\(X = U D V'\).
The resulting class of ellipsoids includes degenerate ellipsoids that are
flat and/or unbounded. Thus ellipsoids are naturally extended to include
lines, hyperplanes, points, cylinders, etc.
The class is closed under linear and affine transformations (including those
between spaces of different dimensions) and under duality ('inverse')
transformations.
Unbounded ellipsoids, e.g. cylinders with elliptical cross-sections,
correspond to singular inner products, i.e. inner products defined by a
singular inner product matrix.
Flat ellipsoids correspond to singular variances. The corresponding inner
product is defined only on the supporting subspace.
Ellipsoids that are both flat and unbounded correspond to lines, points,
subspaces, hyperplanes, etc.
gell can currently generate the U-D representation from 5 ways
of specifying an ellipsoid:
From the non-negative definite dispersion (variance)
matrix, Sigma: \(U D^2 U' = \Sigma\),
where some elements of the diagonal matrix
D can be 0. This can only generate bounded ellipsoids, possibly flat.
From the non-negative definite inner product matrix 'ip': \(U W^2 U = C\)
where some elements of the diagonal matrix W can be 0. Then set D = W^-1
where 0^-1 = Inf. This can only generate fat (non-empty interior)
ellipsoids, possibly unbounded.
From a subspace spanned by 'span' Let U_1 be an orthonormal basis of
Span('span'), let U_2 be an orthonormal basis of the orthogonal complement,
the U = [ U_1 U_2 ] and D = diag( c(Inf,...,Inf, 0,..,0)) where the number
of Inf's is equal to the number of columns of U_1.
From a transformation of the unit sphere given by A(Unit sphere) where
\(A = U D V'\), i.e. the SVD.
(Generalization of 4): (A, d) where A is any matrix and d is a vector of
factors corresponding to columns of A. These factors can be 0, positive or
Inf. In this case U and D are such that U D(Unit sphere) = A diag(d)(Unit
sphere). This is the only representation that can be used for all forms of
ellipsoids and in which any ellipsoid can be represented.