MATLAB: How does matlabs eigs normalise eigenvectors

eigenvalue problemlinear algebranormalisation

In solving the generalised eigenvalue problem Ax=cBx using eig, one gets V and D as outputs where V is the eigenvectors corresponding to the eigenvalues contained in the main diagonal of D. My question is how does matlab normalise these eigenvectors?

In the case of the problem Ax=cx the documentation states 'The eigenvectors in V are normalized so that the 2-norm of each is 1' but for the generalised form 'The 2-norm of each eigenvector is not necessarily 1' (not helpful).

Best Answer

If the matrix B is symmetric positive definite, the eigenvectors are normalized in B-norm (and even orthogonal in B-norm if A is also symmetric). If B is not symmetric positive definite, the 2-norm of each eigenvector is 1, but they will not typically be orthonormal.

Related Solutions

MATLAB: Eig command returning wrong eigenvectors

Doing wrong? First, it looks like you have format set to rat.

[V,D] = eig(K);
V
V =
         0.179605302026751     -5.06266247219048e-14         0.300510985421823        -0.868600674748355      1.49364371455249e-15
                         0                         0                         0                         0         0.757946284651594
        -0.179605302026794         0.324442842261484         0.544556381601175         0.161114353924498         0.166961764835024
         0.898026510133868        -0.811107105653849        -0.444386053127199         -0.45064791217395       -0.0384312697617635
        -0.359210604053568         0.486664263392254         0.644726710075106        -0.128419204324953         0.629415789578334
diag(D)
ans =
          2.00000000000014
          2.99999999999972
          5.00000000000016
                         5
                         3

So eigenvalues of 2,3,5, the latter are of essentially multiplicity 2 each, if we ignore the trash in those least significant bits.

How about the eigenvectors?

You need to recognize that an eigenvector is unique only to within a scale factor. An eigenvector (V) basically has the property

K*V = lambda*V

for corresponding eigenvalue lambda.

But if that holds, then surely

K*(s*V) = lambda*(s*V)

for any scalar value s. So what happens if you scale those eigenvectors?

V(:,1)/V(1,1)
ans =
                       1
                       0
       -1.00000000000024
        5.00000000000064
       -2.00000000000037

How interesting.

So eig is NOT returning the wrong eigenvectors. It merely normalizes the eigenvectors so they have unit norm. Your expectations were wrong.

norm(V(:,1))
ans =
                       1

But that is just scaling the vectors by a constant. And we just discussed that that constant is completely arbitrary. At the same time, having unit normalized eigenvectors is very useful in mathematics.

MATLAB: Norm with matrices help

No, it is not the average.

Assuming you intend the 2-norm, the norm of a matrix is the largest singular value.

Best Answer

Related Solutions

MATLAB: Eig command returning wrong eigenvectors

MATLAB: Norm with matrices help

Related Question