Question

Exercise 5 Use the function matrixTransform to visualise the effect of transformation matrix 4.6 R 2.4 -2.4 Does it behave as you expect? What if you apply the transformation 5 times? Do the points tend towards a particular alignment? Show/hide hint You can also use matrixTransform to visualise rotation matrices, which are of the form RA cos sin e sin cos Below is the code to find transformation matrix for 45 degree rotation >> t = pi/4; >> R45 = [cos(t), -sin(t); sin(t), cos(t)]; Before visualising the rotation in matrixTransform.m, what are the eigenvectors and eigenvalues of the matrix R45? Hint: use eig. Az = Vi = V2 - Now use matrixTransform(R45) to see the transformation. Feel free to try other matrices!

function [Rx] = matrixTransform(R)
% MOD10TRANSFORM is a function which will visualise the effect of your
% 2-by-2 transformation matrix R on a set of grid points. It will also plot
% the eigenvectors of the matrix R if there are two real eigenvectors.

close all;

minX = -10;
maxX = 10;
minY = -10;
maxY = 10;
n = 10;
pad = 5;

corners = [minX, maxX, minX, maxX; maxY, maxY, minY, minY];
corners = [corners, R*corners];
corners = [min(corners(:)) max(corners(:)) min(corners(:)) max(corners(:))] + pad*[-1,1,-1,1];

A = R-eye(2);

[ evec, evals ] = eig(R)

axis(corners);
axis equal;
xLim = get(gca, 'XLim');
yLim = get(gca, 'YLim');

hold on;
if trace(R)^2-4*det(R) > 0
nick = 1;
elseif trace(R)^2-4*det(R) == 0
if isreal(eig(R))
nick = 1;
else
nick = 0;
end
else
nick = 0;
end

for t = linspace(0,1,100)
cla;
if nick
sevec = (A*t+eye(2))*evec;
quiver(0, 0, sevec(1,1)*10, sevec(2,1)*10);
quiver(0, 0, sevec(1,2)*10, sevec(2,2)*10);
end
for x = linspace(minX, maxX, n)
index = 1;
for y = linspace(minY, maxY, n)
if nick
newXY(:,index) = (A*t+eye(2))*[x;y];
  
else
[t1,r1] = cart2pol(x,y);
[t2,r2] = cart2pol(R(1)*x+R(2)*y,R(3)*x+R(4)*y);
if t2 >= t1
t2 = t2-2*pi;
end
[xp, yp] = pol2cart(t1+(t2-t1)*t, r1+(r2-r1)*t);
newXY(:, index) = [xp;yp];
end
index = index+1;
end
plot(newXY(1,:), newXY(2,:), '.', ...
'MarkerEdgeColor',[0.4 (maxX-x)/(maxX-minX) 0.8]);
plot(newXY(1,1), newXY(2,1), '.m');
plot([0, 0],yLim,'-k');
plot(xLim,[0,0],'-k');
end
  
if t == 0
pause(1);
else
pause(0.005);
end
end

end

Answer 1

Here is an implementation in MATLAB.

We have made use of the 'eig' function of MATLAB to find out the eigenvectors and eigenvalues.