how does Matlab normalize generalized eigenvectors?

Question

I know that the eigenvectors produced by eig(A) have 2-norm 1. But what about the vectors produced in the generalized eigenvalue problem eig(A,B)? A natural conjecture is that such a vector v should satisfy v'Bv=1. When B is the identity matrix, then v'Bv is exactly the square of the 2-norm. I ran the following test for various matrices A and B:

[p,d]=eig(A,B);
v=p(:,1);
v'*B*v

I always choose B to be diagonal. I noticed that v'Bv is not always 1. However, it is indeed 1 when A is symmetric. Does anyone know the rule for the way that Matlab normalizes the generalized eigenvectors? I can't find it in the document.

Cris Luengo Cris Luengo · Accepted Answer · 2019-01-11T22:41:40

According to the documentation (emphasis mine):

The form and normalization of V depends on the combination of input arguments:

[...]

[V,D] = eig(A,B) and [V,D] = eig(A,B,algorithm) returns V as a matrix whose columns are the generalized right eigenvectors that satisfy A*V = B*V*D. The 2-norm of each eigenvector is not necessarily 1. In this case, D contains the generalized eigenvalues of the pair, (A,B), along the main diagonal.

When eig uses the 'chol' algorithm with symmetric (Hermitian) A and symmetric (Hermitian) positive definite B, it normalizes the eigenvectors in V so that the B-norm of each is 1.

This means that, unless you are using the 'chol' algorithm, V is not normalized.

how does Matlab normalize generalized eigenvectors?

2 Answers