# qreig.r         
 #
 # compute eigenvalues and vectors 
 #
 # first matrix
   A <- matrix(c(1,3,4,3,2,8,4,8,3),3,3)
   A
   qrA <- eigen(A,symmetric=TRUE)
   qrA
   t(qrA[[2]]) %*% A %*% qrA[[2]]       # diagonal?
 # second matrix
   A <- matrix( c(1,-3,-2,1,-3,10,-3,6,-2,-3,3,-2,1,6,-2,1), 4,4)
   A
   qrA <- eigen(A,symmetric=TRUE)
   qrA
   t(qrA$vectors) %*% A %*% qrA$vectors # diagonal?
 # third matrix
   A <- matrix( c(15,-1,-5,-3,4, -1,12,3,2,3, -5,3,7,-1,2,
         -3,2,-1,9,1, 4,3,2,1,10 ), 5,5)
   A
   qrA <- eigen(A,symmetric=TRUE)
   qrA
   t(qrA[[2]]) %*% A %*% qrA[[2]]       # diagonal?
 # fourth matrix
   A <- matrix( c(8,-3,-2,0,6,5, -3,6,1,-6,0,-2, -2,1,5,-5,2,0,
         0,-6,-5,8,-3,-2, 6,0,2,-3,6,1, 5,-2,0,-2,1,5), 6,6)
   A
   eigen(A,symmetric=TRUE,only.values=TRUE) # eigenvalues only
 # done
   rm(list=ls())
   q()