Calculating separate eigenvectors manually

Hello all,

I am trying to calculate the eigenvectors for the system and I am struggling, the code is as follows:

A=[2 -2i;2i 5];                 % A
lambdaA = round(eig(A));        % Finds values of A
setup1=A-(eye(2)*lambdaA(2,:)); % setting up to produce eigenvectors
Zmatrix=zeros(2,1);             % Zeros
W=linsolve(setup1,Zmatrix)

The could is attempting to set up:

[-4 -2i; 2i -1]*[x;y]=[0;0]

to solve for x and y and place that into W

We know the answer to be:

Any help would be greatly appreciated! Thank you in advance!

Answer 1

[Angelo Yeo]

linsolve cannot provide what you want because where λ is an eigenvalue becomes singular.

Instead, you need to think of how to get the nullspace of . In MATLAB, you can use the function null.

A=[2 -2i;2i 5];                 % 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 = 
   0.0000 + 0.8944i
   0.4472 + 0.0000i

v2 = v2 ./ norm(v2, 2)

v2 = 
   0.0000 - 0.4472i
   0.8944 + 0.0000i

[Christine Tobler]

I agree with Angelo's solution. Just a quick remark to clarify: This is a good way to learn how the eigenvectors and eigenvalues are connected.

In a practical numerical application, it would be preferrable to call [U, D] = eig(A) which returns both the eigenvalues and the eigenvectors.

The "rational" option also makes sense pedagogically, but the standard call to null that uses the svd is preferable numerically.