##
using Random
Random.seed!(0)
A = randn(10,10) |> x -> x + x' # get a random symmetric matrix
10×10 Matrix{Float64}: -0.463818 0.670505 -0.477815 … -1.84353 1.40771 -0.213596 0.670505 1.15601 -0.221631 0.163109 -1.47088 -0.538411 -0.477815 -0.221631 1.90371 -0.550527 1.78206 -1.68827 3.71885 -0.382635 -0.0343197 2.30853 0.160268 0.546554 -0.568353 -1.99942 -0.634999 -0.386993 1.79697 0.898182 3.03331 0.364765 -2.1305 … 0.352134 0.768188 0.102171 -2.27477 -0.741652 -1.14428 -1.73715 -2.03388 0.713051 -1.84353 0.163109 -0.550527 -0.998805 -1.11001 2.38461 1.40771 -1.47088 1.78206 -1.11001 1.03547 -0.988297 -0.213596 -0.538411 -1.68827 2.38461 -0.988297 0.401778
##
using LinearAlgebra
B = A^(10)/opnorm(A^10)
C = (A/opnorm(A))^10
B ≈ C
true
##
C = (A/opnorm(A))^1000
10×10 Matrix{Float64}: 0.251012 0.0257224 0.0662651 0.186256 … 0.182446 -0.0606247 0.0257224 0.00263589 0.0067905 0.0190865 0.0186961 -0.0062125 0.0662651 0.0067905 0.0174934 0.04917 0.0481644 -0.0160044 0.186256 0.0190865 0.04917 0.138205 0.135379 -0.0449847 -0.0200418 -0.00205377 -0.00529086 -0.0148714 -0.0145672 0.00484051 0.128662 0.0131846 0.0339657 0.0954697 … 0.0935172 -0.0310746 -0.306483 -0.0314067 -0.0809089 -0.227416 -0.222765 0.074022 0.0202952 0.00207974 0.00535776 0.0150594 0.0147514 -0.00490172 0.182446 0.0186961 0.0481644 0.135379 0.13261 -0.0440647 -0.0606247 -0.0062125 -0.0160044 -0.0449847 -0.0440647 0.0146421
##
x = randn(10)
y = C*x / norm(C*x)
z = randn(10)
w = C*z / norm(C*z)
y ≈ w
false
##
C ≈ y*y'
true