Dual or 'Inverse' of an ellipsoid

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 class 'gell'
dual(x, ...)

Arguments

x: An object, of class "gell"
...: Other arguments, unused for now.

Value

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

u: Right singular vectors
d: Singular values

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.

Author

Georges Monette

Examples


(zplane <- gell(span = diag(3)[,1:2]))  # a plane
#> $center
#> [1] 0 0 0
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> $d
#> [1] Inf Inf   0
#> 
#> attr(,"class")
#> [1] "gell"

dual(zplane)  # line orthogonal to that plane
#> $center
#> [1] 0 0 0
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    0    1    0
#> [3,]    1    0    0
#> 
#> $d
#> [1] Inf   0   0
#> 
#> attr(,"class")
#> [1] "gell"

(zhplane <- gell(center = c(0,0,2), span = diag(3)[,1:2]))  # a hyperplane
#> $center
#> [1] 0 0 2
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> $d
#> [1] Inf Inf   0
#> 
#> attr(,"class")
#> [1] "gell"

dual(zhplane) # orthogonal line through same center (note that the 'gell'
#> $center
#> [1] 0 0 2
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    0    1    0
#> [3,]    1    0    0
#> 
#> $d
#> [1] Inf   0   0
#> 
#> attr(,"class")
#> [1] "gell"
              # object with a center contains more information than the geometric plane)

zorigin <- gell(span = cbind(c(0,0,0)))
dual( zorigin )
#> $center
#> [1] 0 0 0
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    0    1    0
#> [3,]    1    0    0
#> 
#> $d
#> [1] Inf Inf Inf
#> 
#> attr(,"class")
#> [1] "gell"