What do the components of eigenvector represent?

When I use [V,D] = eig(A), V has 2 X 2 matrix. What do the components of V represent? V(1,1) represents the angle between x-axis and principal direction, and V(2,1) represents the angle between y-axis and principal direction?? I would appreciate any help. Thanks,

Jay

Answer 1

[Youssef Khmou]

N dimensional matrix is associated with N dimensional canonical base, in this case N=2, you have a plane (x,y), after eigendecomposition you have the diagonal matrix D which contains the spectra of the matrix A and the columns of V are the associated eigenvectors V(:,1)= V1 ex +V2 ey such as V(1,1) and V(2,1) are the x and y components of the first eigenvector .

 V(1,1)= ||V1|| cos(theta)
 V(2,1)= ||V1|| sin(theta)

[Jay]

I used 'acosd' and 'asind' to check if the V components are representing the x and y components. For some cases, they are right, but not for all. I got V from one of the matrices which contains [-0.0615;-0.9981]. I calculated theta and I got 93.5259 and -86.4675 degree respectively.

When I use atan2(-0.0615,-0.9981), I got -176.74 degree. Which is correct? Sorry that I might not fully understand your answer. I would appreciate any help.

[Youssef Khmou]

The correct answer is -93.5259, there is ambiguity is sign (cos(theta)=cos(-theta) :

 p=[-0.0615;-0.9981];
 theta=-acosd(dot([1 0],p))

You can verify this as p is normalized |p|=1 :

 cosd(theta)
 sind(theta)

[Jay]

Thanks, Youssef. I would like to confirm if I am doing correctly. When I get D= [D1, 0 ; 0, D4] and V = [V1, V2; V3, V4], and if D1 > D4, D1 is the first principal sth and I need to use V1 and V3 as the principal direction. Am I correct?

[Youssef Khmou]

Yes correct, when you use 'eig', the diagonal elements Dii are sorted in descending order.

[Jay]

What do you mean descending order? D11 is always greater than D22? In my model, sometimes D22 is greater.

[Youssef Khmou]

check this example : eig(randn(10)), can you post a counter example?

[Jay]

B =

    0.2696    0.4800
    0.4800   -1.0203

>> [V,D]=eig(B)

V =

   -0.3145   -0.9493
    0.9493   -0.3145

D =

   -1.1793         0
         0    0.4287

[Jay]

I am trying to draw principal directions on a plot.

% these codes are in for loop

[V,D] = eig(e2_j_k);

        if (D(1,1) > D(2,2))
           eigen_j_k = D(1,1);
           p= [V(1,1);V(2,1)];
           if (V(2,1) < 0)
               theta_j_k = -acosd(dot([1 0],p));
           else
               theta_j_k = acosd(dot([1 0],p));
           end

        else
           eigen_j_k = D(2,2);
           p= [V(1,2);V(2,2)];
           if (V(2,2) < 0)
               theta_j_k = -acosd(dot([1 0],p));
           else
               theta_j_k = acosd(dot([1 0],p));
           end

eval(sprintf('eigen_%d_%d = eigen_j_k',j,k));
eval(sprintf('theta_%d_%d = theta_j_k',j,k));

% I already know xc and zc

xc2(j,k) = xc(j,k) + 0.3*cosd(theta(j,k));
zc2(j,k) = zc(j,k) + 0.3*sind(theta(j,k));

figure(6);
plot([xc(j,k),xc2(j,k)],[zc(j,k),zc2(j,k)]);
hold on;

The code above is that I am trying to draw the directions of principal strain. I don't get any reasonable results. Do you find anything wrong or would you have other ways to draw? Any ideas? I would appreciate any help.

Jay