%
% Graphical illustration of matrix norms, 16.2.2022
%
% input: 
%   A - 2x2 matrix with real nonzero eigenvalues
%   inorm - indicator of the used norm:
%      1 = column, 2 = Eucliedan, 'inf' = row
%
% output:
%   fig. 1 - before transformation:
%           unit "circle" (blue)
%           eigenvectors (red)
%   fig. 2 - after transformation
%           radius of the dashed circle is spectral radius of A
%           "radius" of the green "circle" is norm of A


close all
clear all

inorm = 2; % 1 = column, 2 = Eucliedan, 'inf' = row

 A = [1 0.5; 2 -1.5];  % general transformation, real eigenv.
% A = [ 3 -1; -1 2];    % symmetric transformation
% A = [ 0.9 0.6; 0 0.6];    % real eigenv., ro < 1, norm > 1
% A = [ 1 1; 0 1];      % shear
% A = [1 -1; 1 2];      % general transformation, imag. eigenv.


  % X - unit "circle" (depends on norm)
if inorm == 2
  t = linspace(0,2*pi,40);
  X = [sin(t);cos(t)];
elseif inorm == 1
  X = [1 0 -1 0 1; 0 1 0 -1 0];
else
  X = [-1 1 1 -1 -1; -1 -1 1 1 -1];
end
plot(X(1,:),X(2,:),'linewidth',2)
axis equal;
axis([ -1.3  1.3  -1.1  1.1 ])
hold on;

  % V - eigenvectors, E - eigenvalues of A
[V,E] = eig(A);
realV = isreal(V);
if realV
    % plot the (unit norm) eigenvectors
  V1 = V(:,1); V2 = V(:,2);
  d1 = norm(V1,inorm); d2 = norm(V2,inorm); 
  UP = [ [0;0] V1/d1,  [0;0] V2/d2 ];
  plot(UP(1,:),UP(2,:),'r','linewidth',2)
else
  disp('eigenvectors are imaginary')
end

figure(2);

  % unit "circle" 
plot(X(1,:),X(2,:), '--','linewidth',2)
axis equal;
hold on

  % "circle" of the size of the given matrix norm
nr = norm(A, inorm);
plot(nr*X(1,:),nr*X(2,:), 'g','linewidth',2)
plot([nr,0], [0,0], 'g','linewidth',2)

  % spectral radius "circle"
disp('eigenvalues:');
disp(num2str(diag(E)));
s = max(abs(diag(E)));
plot(s*X(1,:),s*X(2,:), '--k','linewidth',2) 
plot([s,0], [0,0], '--k','linewidth',2)

  % transformation of the unit "circle"
B = A*X;
plot(B(1,:),B(2,:),'linewidth',2)

if realV
    % plot transformation of the eigenvectors
  UP = A*UP;
  plot(UP(1,:),UP(2,:),'r','linewidth',2)
end

%%%%%%%%%%%%%%%%%%%%%%%%%