How to calculate eigenvectors without using eig

I have a matrix, I need to get the eigenvectors. I already calculated the eigenvalues, Let's assume we have the eigenvalues, I wrote this

 for i = 1:length(c)
  
syms y
cal_vec = (c-eig_Val(i)*I)*y == 0;
  
eigVec(:,i) = double(solve(cal_vec,y));

 end

now I got zero as y, but I need to get y 1 and y2

Answer 1

[Matt J]

Hint: use the null command to find non-zero solutions to the eigenvector equation.

[IDRIS Badmus]

can you explain further please, i don't understand what you mean

[Matt J]

There is a Matlab command called null (documentation here) which will find vectors in the null space of a matrix.

[Tyler Bilheimer]

I dont understand where you're even supposed to put null in this

[Matt J]

Well, the eigenvectors are by definition the null vectors of the matrix , so it should be straightforward to build that matrix and apply null() to it.

Answer 2

[Angelo Yeo]

Although this question is getting old, here is a sample solution to the question.

A=[2 1; 1, 2];                 % A
lambdaA = round(eig(A));        % Finds values of A
% Note that "rational" option is used otherwise SVD is used in the
% calculation. 
v1 = null(A - lambdaA(1) * eye(2), "rational");
v2 = null(A - lambdaA(2) * eye(2), "rational");

v1 = v1 ./ norm(v1, 2)

v1 = 2×1

-0.7071 0.7071

v2 = v2 ./ norm(v2, 2)

v2 = 2×1

0.7071 0.7071

[Syed Abdullah]

You're still using eig builtin function.

[Steven Lord]

A=[2 1; 1, 2];                 % A
lambdaA = [1, 3]; % Eigenvalues calculated earlier
% Note that "rational" option is used otherwise SVD is used in the
% calculation. 
v1 = null(A - lambdaA(1) * eye(2), "rational");
v2 = null(A - lambdaA(2) * eye(2), "rational");

v1 = v1 ./ norm(v1, 2)

v1 = 2×1

-0.7071 0.7071

v2 = v2 ./ norm(v2, 2)

v2 = 2×1

0.7071 0.7071

Now, to check v1 and v2, let's call eig and compare the result of the code above with the "known" answer.

[V, D] = eig(A)

V = 2×2

-0.7071 0.7071 0.7071 0.7071

D = 2×2

1 0 0 3

That looks good to me.

[Walter Roberson]

The question is about calculation of eigenvectors knowing the eigenvalues