MATLAB: Right eigenvector corresponding to an eigenvalue 1

eigenvaluematricesmatrix

How can I find the eigenvector corresponding to the eigenvalue

Best Answer

As the documentation for the eig function says:

[V,D] = eig(A) returns diagonal matrix D of eigenvalues and matrix V whose columns are the corresponding right eigenvectors, so that A*V = V*D.

So, if you use the command above, for example

D(2,2) would give you the 2nd eigenvalue
V(:,2) would give you the 2nd eigenvector

- Sebastian

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MATLAB: Eigenvector different to expected using eig

A=[1.5 -0.5; -1 1];
[Vec,Val]=eig(A)
Vec = 2×2
    0.7071    0.4472
   -0.7071    0.8944
Val = 2×2
    2.0000         0
         0    0.5000

What does the eig function return? From its documentation: "[V,D] = eig(A) returns diagonal matrix D of eigenvalues and matrix V whose columns are the corresponding right eigenvectors, so that A*V = V*D." So is A*V close to V*D?

format longg
A*Vec-Vec*Val
ans = 2×2
                         0      2.77555756156289e-17
                         0                         0

Yeah, those are pretty close to 0. How about for your matrix of eigenvectors?

Vec2 = [1 1; 2 -1]
Vec2 = 2×2
     1     1
     2    -1
A*Vec2-Vec2*Val
ans = 2×2
                      -1.5                       1.5
                        -3                      -1.5

Perhaps, from the orientation of your eigenvectors, you're trying to use the left eigenvectors? "[V,D,W] = eig(A) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'."

[Vec, Val, VecLeft] = eig(A);
VecLeft
VecLeft = 2×2
         0.894427190999916         0.707106781186547
        -0.447213595499958         0.707106781186547
VecLeft'*A - Val*VecLeft'
ans = 2×2
     0     0
     0     0
Vec2'*A-Val*Vec2'
ans = 2×2
                      -2.5                      -2.5
                         2                        -1

Still no. So please show me why you believe that the columns of Vec2 are eigenvectors for this matrix.

If we scaled VecLeft a bit it looks a little similar to your Vec2 matrix but not exactly.

VecLeftScaled = VecLeft ./ VecLeft(2, :)
VecLeftScaled = 2×2
    -2     1
     1     1
VecLeftScaled'*A-Val*VecLeftScaled'
ans = 2×2
     0     0
     0     0

MATLAB: How to find lambda from this formula

e = eig(A) returns a vector of eigenvalues of the matrix A. To get the associated eigenvectors, use [V,D] = eig(A). See the documentation for eig to interpret V and D.

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