8.2. Eigenvector Animation

Eigenvectors and eigenvalues can be difficult to understand, so the MATLAB code below shows an animation that will hopefully help to visualize what makes a vector an eigenvector. The code defines a function that plots an animation when a 2x2 matrix is passed to it. The matrix must have real eigenvectors and eigenvalues, which is always achieved when the matrix is symmetric (equal to its own transpose).

More information is in the comments of the function.

function eigdemo(A)
%% Eigenvector, eigenvalue Demo for input 2x2 matrix.
%
%  When eigenvectors of A are mulitiplied by A, the result is a scalar
%  multiple of the vector. The scalar values (lambda) are the eigenvalues
%  of A. It is the orientation of the eigenvectors that makes them special.
%
%  A*x == lambda*x,  C*A*x == C*lambda*x
%
%  This function shows an animation demonstrating that the product (A*x) is
%  inline with the vector x when, and only when, x is an eigenvector. The A
%  (input) matrix must be a 2x2 matrix with real eigenvalues.
%
%  The red rotating line shows possible unit vectors, x.
%  The blue moving line is v = A*x.
%  The green fixed arrows are the product (lamda*eigenvectors == A*ex),
%
%  Notice that when the red rotating vector is inline win an eigenvector
%  that the blue line matches the green arrows. The blue line is also (-1)
%  of the green arrow when the red rotating line is (-1) of an eigenvector.
%


if size(A,1) ~= 2 || size(A,2) ~= 2
    disp('Matrix A must be 2x2')
    return;
end

theta = linspace(2*pi,0,100);
xv = cos(theta);
yv = sin(theta);
v = A*[xv; yv];
[ex, lambda] = eig(A);
if ~isreal(ex) || ~isreal(lambda)
    disp('Complex Eigenvector results, use another A matrix');
    return;
end
ep = A*ex;       % same as A*ex == lambda*ev
fprintf('eigenvector 1: %f %f\n', ex(1,1), ex(2,1));
fprintf('eigenvector 2: %f %f\n', ex(1,2), ex(2,2));
fprintf('A*eigenvector 1: %f %f\n', ep(1,1), ep(2,1));
fprintf('A*eigenvector 2: %f %f\n', ep(1,2), ep(2,2));
fprintf('lambda: %f %f\n', lambda(1,1), lambda(2,2));

M = ceil(max(vecnorm(v)));
figure; axis([-M M -M M]);
daspect([1 1 1]);
hold on
l1 = line([0 1], [0 0], 'LineWidth', 3, 'Color', 'r');
l2 = line([0 1], [0 0], 'LineWidth', 3, 'Color', 'b');
plot2vector(ep(:,1), 'g', 2); % lamda 1 * eigenvector 1
plot2vector(ep(:,2), 'g', 2); % lamda 2 * eigenvector 2
hold off

for k = 1:3        % make three loops
    for i = 1:100
        l1.set('XData', [0 xv(i)], 'YData', [0 yv(i)]);
        l2.set('XData', [0 v(1,i)], 'YData', [0 v(2,i)]);
        drawnow;
        pause(0.1);
    end
end
end

function plot2vector(x, color, LineWidth)
    if iscolumn(x)
        x = x';
    end
    plot([0, x(1)], [0, x(2)], 'Color', color, 'LineWidth', LineWidth)
end