gellipsoid (version 0.7.3)

dual: Dual or 'Inverse' of an ellipsoid

Description

dual produces the orthogonal complement for subspaces or for ellipsoids. This is equivalent to inverting \(\Sigma\) or an inner product ip when these are non-singular.

Usage

dual(x, ...)
# S3 method for gell
dual(x, ...)

Value

A (U, D) representation of the dual, with components

LIST, use

u

Right singular vectors

d

Singular values

Arguments

x

An object, of class "gell"

...

Other arguments, unused for now.

Author

Georges Monette

Details

At present, dual is only defined for objects of class "gell".

In the (U,D) representation, the dual simply has the columns of U in the reverse order, and the reciprocals of the diagonal elements of D, also in reverse order.

References

Dempster, A. (1969). Elements of Continuous Multivariate Analysis Reading, MA: Addison-Wesley.

See Also

gell

Examples

(zplane <- gell(span = diag(3)[,1:2]))  # a plane

dual(zplane)  # line orthogonal to that plane

(zhplane <- gell(center = c(0,0,2), span = diag(3)[,1:2]))  # a hyperplane

dual(zhplane) # orthogonal line through same center (note that the 'gell'
              # object with a center contains more information than the geometric plane)

zorigin <- gell(span = cbind(c(0,0,0)))
dual( zorigin )