eigenvalues of a shift-average operation on a circle

circle_shift.R

# first page of www.math.sunysb.edu/~kirillov/mat552/liegroups.pdf

# However, explicitly computing the characteristic polynomial and finding the roots is a rather difficult problem.


#He must mean for general N because this seems to work for me??


require(magrittr)
require(matrixcalc)
#result of much futzing
{diag(5) %>% function(x) shift.left(x) + shift.right(x) %>% function(x) x+ rbind( c(rep(0,4),1), rep(0,5), rep(0,5), rep(0,5), c(1,rep(0,4)) ) } -> shape



#> shape
#     [,1] [,2] [,3] [,4] [,5]
#[1,]    0    1    0    0    1
#[2,]    1    0    1    0    0
#[3,]    0    1    0    1    0
#[4,]    0    0    1    0    1
#[5,]    1    0    0    1    0



# shape/2 is the desired matrix for shifting 5 points on a circle as described in the PDF




N=55
{diag(N) %>% function(x) shift.left(x,fill=c(1,rep(0,N-1)) ) + shift.right(x,fill=c(rep(0,N-1),1) )} -> big

eigen(big/2)