get_eigproj.spectral {qwalkr}

Extract an Eigen-Projector from a Hermitian Matrix

Description

Get the orthogonal projector associated with an eigenspace based on the representation of a Hermitian Matrix given by class spectral.

Usage

## S3 method for class '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}^*

See Also

spectral(), get_eigproj()

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)

# Returns the projector associated to the eigenvalue 2.
get_eigproj(decomp, id=1)