I see what you mean, but say you have such a matrix and there are two eigenvalues that you can't separate from each other. Then wouldn't their corresponding eig

I see what you mean, but say you have such a matrix and there are two eigenvalues that you can't separate from each other. Then wouldn't their corresponding eigenvectors define like a 2d plane? Couldn't there still be some interval for that plane, like say that the plane is defined by the linear combination of two vectors and both vectors have zero on some coordinate, say the x coordinate. The interval could still be helpful by saying that any combination will also be zero there, right?