I see you've reworded "independent dimensions" to "independent variables". This indicates to me a lacuna in your understanding of eigenvectors, possibly because symmetric matrices are the only matrices you are intimately familiar with.
You should consider what happens with rotation matrices (no real eigenvectors), defective matrices (no complete set of eigenvectors), and the Jordan form. Wikipedia has adequate articles on these topics.
Also, about the Jordan form, if you've only ever worked with matrices over floats instead of exact precision, it would be understandable why you are not familiar with the Jordan form, as it's numerically useless, because it's numerically unstable. The tiniest perturbation will change a defective matrix into a diagonalisable matrix.
You should consider what happens with rotation matrices (no real eigenvectors), defective matrices (no complete set of eigenvectors), and the Jordan form. Wikipedia has adequate articles on these topics.
Also, about the Jordan form, if you've only ever worked with matrices over floats instead of exact precision, it would be understandable why you are not familiar with the Jordan form, as it's numerically useless, because it's numerically unstable. The tiniest perturbation will change a defective matrix into a diagonalisable matrix.