Eigenvectors not changing with constant parameter

Dear All,

I am trying to calculate the eigenvectors(V) using eig() function. My 2x2 matrix(M) contains a constant parameter 'a' in it. But I see that changing a does not change my eigenvectors. Generally the eigevectors and eigenvalues change with the matrix elements. Here my eigenvalues are varying but not the eigenvectors. Could someone figure out the issue?

sx = [0 1; 1 0];
sy = [0 -1i; 1i 0];
a = 0.18851786;
kx = -0.5:0.1:0.5;
ky = kx;
for i = 1:length(kx)
    for j = 1:length(ky)
        M = a.*(sx.*ky(i)-sy.*kx(j));
        [V,D] = eig(M);
        V
    end
end

Answer 1

[Bruno Luong]

"Could someone figure out the issue?"

But there is no issue beside thet fact that you expect something that not going to happen.

if V and diagonal D the eigen decomposition of A1

A1*V = V*D

then for any constant a

(a*A1)*V = V*(a*D)

Meaning V and a*D (still diagonal) are eigen decomposition of Aa := a*A1.

So A1 and Aa respective eigen decomposition can have the same V (eigen vectors, MATALB always normalized them to have norm(V(:,k),2)=1 for all k) but eigen values are proportional to a.

[SHUBHAM PATEL]

Thanks Bruno,

It is right that multiplying the whole matrix with a constant does not have any effect on eigenvectors.

My point is if I change the off diagonal term by a factor (a), then it should change the eigenvectors, right?

Could you please check the comment section of Walter's reply. I wrote the code again there.

Thanks again.

[Bruno Luong]

Look like they change, so what is your point?

sx = [0 1; 1 0];
sy = [0 -1i; 1i 0];
I = eye(2);
a = 0.18851786;
kx = -0.5:0.1:0.5;
ky = kx;
for i = 1:length(kx)
    for j = 1:length(ky)
        M = (kx(i).^2+ky(j).^2)*I + a.*(sx.*ky(i)-sy.*kx(j));
        [V,D] = eig(M);
        V
    end
end

V = 
  -0.5000 - 0.5000i   0.5000 + 0.5000i
  -0.7071 + 0.0000i  -0.7071 + 0.0000i

V = 
   0.5522 + 0.4417i  -0.5522 - 0.4417i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
  -0.6063 - 0.3638i   0.6063 + 0.3638i
  -0.7071 + 0.0000i  -0.7071 + 0.0000i

V = 
   0.6565 + 0.2626i  -0.6565 - 0.2626i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6934 + 0.1387i  -0.6934 - 0.1387i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 -0.7071 -0.7071 0.7071

V = 
   0.6934 - 0.1387i  -0.6934 + 0.1387i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6565 - 0.2626i  -0.6565 + 0.2626i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
  -0.6063 + 0.3638i   0.6063 - 0.3638i
  -0.7071 + 0.0000i  -0.7071 + 0.0000i

V = 
   0.5522 - 0.4417i  -0.5522 + 0.4417i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
  -0.5000 + 0.5000i   0.5000 - 0.5000i
  -0.7071 + 0.0000i  -0.7071 + 0.0000i

V = 
   0.4417 + 0.5522i  -0.4417 - 0.5522i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i  -0.5000 - 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5657 + 0.4243i  -0.5657 - 0.4243i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6325 + 0.3162i  -0.6325 - 0.3162i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6860 + 0.1715i  -0.6860 - 0.1715i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 -0.7071 -0.7071 0.7071

V = 
   0.6860 - 0.1715i  -0.6860 + 0.1715i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6325 - 0.3162i  -0.6325 + 0.3162i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5657 - 0.4243i  -0.5657 + 0.4243i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i  -0.5000 + 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.4417 - 0.5522i  -0.4417 + 0.5522i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3638 + 0.6063i  -0.3638 - 0.6063i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.4243 + 0.5657i  -0.4243 - 0.5657i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i  -0.5000 - 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5883 + 0.3922i  -0.5883 - 0.3922i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6708 + 0.2236i  -0.6708 - 0.2236i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 -0.7071 -0.7071 0.7071

V = 
   0.6708 - 0.2236i  -0.6708 + 0.2236i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5883 - 0.3922i  -0.5883 + 0.3922i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i  -0.5000 + 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.4243 - 0.5657i  -0.4243 + 0.5657i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3638 - 0.6063i  -0.3638 + 0.6063i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2626 + 0.6565i  -0.2626 - 0.6565i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3162 + 0.6325i  -0.3162 - 0.6325i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
  -0.3922 - 0.5883i   0.3922 + 0.5883i
  -0.7071 + 0.0000i  -0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i  -0.5000 - 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6325 + 0.3162i  -0.6325 - 0.3162i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 -0.7071 -0.7071 0.7071

V = 
   0.6325 - 0.3162i  -0.6325 + 0.3162i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i  -0.5000 + 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
  -0.3922 + 0.5883i   0.3922 - 0.5883i
  -0.7071 + 0.0000i  -0.7071 + 0.0000i

V = 
   0.3162 - 0.6325i  -0.3162 + 0.6325i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2626 - 0.6565i  -0.2626 + 0.6565i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1387 + 0.6934i  -0.1387 - 0.6934i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1715 + 0.6860i  -0.1715 - 0.6860i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2236 + 0.6708i  -0.2236 - 0.6708i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3162 + 0.6325i  -0.3162 - 0.6325i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i  -0.5000 - 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 -0.7071 -0.7071 0.7071

V = 
   0.5000 - 0.5000i  -0.5000 + 0.5000i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3162 - 0.6325i  -0.3162 + 0.6325i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2236 - 0.6708i  -0.2236 + 0.6708i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1715 - 0.6860i  -0.1715 + 0.6860i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1387 - 0.6934i  -0.1387 + 0.6934i
   0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 - 0.7071i   0.0000 - 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 - 0.7071i   0.0000 - 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 - 0.7071i   0.0000 - 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 - 0.7071i   0.0000 - 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 - 0.7071i   0.0000 - 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

1 0 0 1

V = 
   0.0000 + 0.7071i   0.0000 + 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 + 0.7071i   0.0000 + 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 + 0.7071i   0.0000 + 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 + 0.7071i   0.0000 + 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.0000 + 0.7071i   0.0000 + 0.7071i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1387 - 0.6934i   0.1387 - 0.6934i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1715 - 0.6860i   0.1715 - 0.6860i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2236 - 0.6708i   0.2236 - 0.6708i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3162 - 0.6325i   0.3162 - 0.6325i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i   0.5000 - 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 0.7071 0.7071 0.7071

V = 
   0.5000 + 0.5000i   0.5000 + 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3162 + 0.6325i   0.3162 + 0.6325i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2236 + 0.6708i   0.2236 + 0.6708i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1715 + 0.6860i   0.1715 + 0.6860i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.1387 + 0.6934i   0.1387 + 0.6934i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2626 - 0.6565i   0.2626 - 0.6565i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3162 - 0.6325i   0.3162 - 0.6325i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3922 - 0.5883i   0.3922 - 0.5883i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i   0.5000 - 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6325 - 0.3162i   0.6325 - 0.3162i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 0.7071 0.7071 0.7071

V = 
   0.6325 + 0.3162i   0.6325 + 0.3162i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i   0.5000 + 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3922 + 0.5883i   0.3922 + 0.5883i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3162 + 0.6325i   0.3162 + 0.6325i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.2626 + 0.6565i   0.2626 + 0.6565i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3638 - 0.6063i   0.3638 - 0.6063i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.4243 - 0.5657i   0.4243 - 0.5657i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i   0.5000 - 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5883 - 0.3922i   0.5883 - 0.3922i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6708 - 0.2236i   0.6708 - 0.2236i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 0.7071 0.7071 0.7071

V = 
   0.6708 + 0.2236i   0.6708 + 0.2236i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5883 + 0.3922i   0.5883 + 0.3922i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i   0.5000 + 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.4243 + 0.5657i   0.4243 + 0.5657i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.3638 + 0.6063i   0.3638 + 0.6063i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.4417 - 0.5522i   0.4417 - 0.5522i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i   0.5000 - 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5657 - 0.4243i   0.5657 - 0.4243i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6325 - 0.3162i   0.6325 - 0.3162i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6860 - 0.1715i   0.6860 - 0.1715i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 0.7071 0.7071 0.7071

V = 
   0.6860 + 0.1715i   0.6860 + 0.1715i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6325 + 0.3162i   0.6325 + 0.3162i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5657 + 0.4243i   0.5657 + 0.4243i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i   0.5000 + 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.4417 + 0.5522i   0.4417 + 0.5522i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 - 0.5000i   0.5000 - 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5522 - 0.4417i   0.5522 - 0.4417i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6063 - 0.3638i   0.6063 - 0.3638i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6565 - 0.2626i   0.6565 - 0.2626i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6934 - 0.1387i   0.6934 - 0.1387i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 2×2

-0.7071 0.7071 0.7071 0.7071

V = 
   0.6934 + 0.1387i   0.6934 + 0.1387i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6565 + 0.2626i   0.6565 + 0.2626i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.6063 + 0.3638i   0.6063 + 0.3638i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5522 + 0.4417i   0.5522 + 0.4417i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

V = 
   0.5000 + 0.5000i   0.5000 + 0.5000i
  -0.7071 + 0.0000i   0.7071 + 0.0000i

[SHUBHAM PATEL]

Now if you change the value of 'a',the vectors should change. It is not happening. Please check.

[Bruno Luong]

This is just a coincidence in YOUR parametrization.

Your matrix M has the same eigen vectors as this matrix

A := M / (kx(i).^2+ky(j).^2) % see your original question
A = I + K(a);

with

K = [0 conj(z);
     z 0]
z(a) := a*(kx(i) + 1i*ky(j)) / (kx(i).^2+ky(j)).^2 % but it doesn't matter, any complex still works

So

A = [1 conj(z);
     z 1];

This (Hermitian) matrix A has two eigen values and (unormalized) eigen vectors

d1 = 1 - abs(z); V1 = [conj(z); -abs(z)];
d2 = 1 + abs(z); V2 = [conj(z); +abs(z)];

So when you scale z up and down by a (a non zero real value) the vector V1 and V2 remains the same after normalization (by MATLAB.

V1 = [conj(z); -abs(z)] / sqrt(2*abs(z)^2) = [conj(z)/abs(z), -1] / sqrt(2)
V2 = [conj(z);  abs(z)] / sqrt(2*abs(z)^2) = [conj(z)/abs(z), +1] / sqrt(2)

Quod erat demonstrandum

[SHUBHAM PATEL]

Thank you @Bruno Luong for such a detailed explaination. I understood the gist.

Answer 2

[Walter Roberson]

eigenvectors are geometrically directions. When you scale a matrix by a nonzero constant, the direction does not change.

[SHUBHAM PATEL]

Thanks W. Roberson,

I understood your point that if we multiply the matrix by a constant it does not change the directions.

But I am seeing that if I include the diagonal term also and multiply the off-diagonal term only with a constant, even then the vectors are not changing, although the eigen values are changing.

Could you comment on that?

I am writing the code again including the diagonal term.

sx = [0 1; 1 0];
sy = [0 -1i; 1i 0];
a = 0.18851786;
kx = -0.5:0.1:0.5;
ky = kx;
for i = 1:length(kx)
    for j = 1:length(ky)
        M = (kx(i).^2+*ky(j).^2)*I + a.*(sx.*ky(i)-sy.*kx(j));
        [V,D] = eig(M);
        V
    end
end