Get the orthogonal projector associated with an eigenspace based on the representation
of a Hermitian Matrix given by class spectral.
Usage
# S3 method for spectral
get_eigproj(object, id, ...)Arguments
- object
an instance of class
spectral.- id
index for the desired eigenspace according to the ordered (decreasing) spectra.
- ...
further arguments passed to or from other methods.
Value
The orthogonal projector of the desired eigenspace.
A Hermitian matrix S admits the spectral decomposition \(S = \sum_{r}\lambda_r E_r\)
such that \(E_r\) is the orthogonal projector onto the \(\lambda_r\)-eigenspace. If \(V_{id}\)
is the matrix associated to the eigenspace, then
$$E_{id} = V_{id}V_{id}^*$$
Examples
# Spectra is {2, -1} with multiplicities one and two respectively.
decomp <- spectral(matrix(c(0,1,1,1,0,1,1,1,0), nrow=3))
# Returns the projector associated to the eigenvalue -1.
get_eigproj(decomp, id=2)
#> [,1] [,2] [,3]
#> [1,] 0.6666667 -0.3333333 -0.3333333
#> [2,] -0.3333333 0.6666667 -0.3333333
#> [3,] -0.3333333 -0.3333333 0.6666667
# Returns the projector associated to the eigenvalue 2.
get_eigproj(decomp, id=1)
#> [,1] [,2] [,3]
#> [1,] 0.3333333 0.3333333 0.3333333
#> [2,] 0.3333333 0.3333333 0.3333333
#> [3,] 0.3333333 0.3333333 0.3333333