act_eigfun.spectral: Apply a Function to a Hermitian Matrix

In qwalkr: Handle Continuous-Time Quantum Walks with R

Apply a Function to a Hermitian Matrix

Description

Apply a function to a Hermitian matrix based on the representation given by class spectral.

Usage

## S3 method for class 'spectral'
act_eigfun(object, FUN, ...)

Arguments

object	an instance of class spectral.
FUN	the function to be applied to the matrix.
...	further arguments passed on to FUN.

Value

The matrix resulting from the application of FUN.

A Hermitian Matrix admits the spectral decomposition

H = \sum_k \lambda_k E_k

where \lambda_k are its eigenvalues and E_k the orthogonal projector onto the \lambda_k-eigenspace.

If f=FUN is defined on the eigenvalues of H, then act_eigfun performs the following calculation

f(H) = \sum_k f(\lambda_k) E_k

See Also

spectral(), act_eigfun()

Examples

H <- matrix(c(0,1,1,1,0,1,1,1,0), nrow=3)
decomp <- spectral(H)

# Calculates H^2.
act_eigfun(decomp, FUN = function(x) x^2)

# Calculates sin(H).
act_eigfun(decomp, FUN = function(x) sin(x))

# Calculates H^3.
act_eigfun(decomp, FUN = function(x, y) x^y, 3)

qwalkr documentation built on Sept. 27, 2023, 9:09 a.m.