# Linear algebra of operators

1. Sep 17, 2006

### JohanL

If you take the outer product of two orthogonal 4-dimensional vectors you get a 4*4 matrix. I wonder what you generally can say about this matrix eigenvectors? For example: If 2,3 or 4 of the eigenvalues are zero is it always true that the matrix have at least 2 linear dependent eigenvectors? Maybe this is true regardless of the values of the eigenvalues?

The physical problem is about the operator |a><b| where |a> and |b> are orthogonal to each other and the characteristic polynom is x^4. I have the values for this operator in a basis and it can not be diagonalized.
Is this maybe a general property when |a> and |b> are orthogonal? Or when a couple of the eigenvalues are zero?
If so what is the physics behind this?

Any ideas?