%%
clear
clc

%% 
% Let's start with a known set of eigenvalues and eigenvectors
D = diag([-6 -1 2 4 5]);
[Q, R] = qr(randn(5));
A = Q*D*Q'

%%
eig(A)

%%
% Now we will take the QR factorization and just reverse the factors.
[Q, R] = qr(A);
A = R*Q

%%
% It turns out that this is a similarity transformation, so the eigenvalues
% are unchanged.
eig(A)

%%
% What's remarkable is that if we repeat the transformation many times, the
% process converges to D. 
for k = 1:15
    [Q, R] = qr(A);
    A = R*Q;
end
A