Is unit vector an eigenvector

Question

From my understanding an Eigenvector is a vector that does not change during a transformation, it just scales. Intuitively, it feels that a unit vector should never change during a transformation, is that true?

I.e. is a unit always an eigenvector?

Answer 1

From definition for a real eigenvalue $\lambda$ we have

$$A\vec x=\lambda \vec x$$

then the (an) eigenvector $\vec x$ is scaled by a factor $\lambda$ independentely from its norm/length.

Notably if we consider the normalized eigenvector $\frac{\vec x}{|\vec x|}$ we have indeed that

$$A\frac{\vec x}{|\vec x|}=\lambda \frac{\vec x}{|\vec x|}$$

Note also that in the general case of complex eigenvalues the direction could not be preserved.

Answer 2

Depending on transformation. For example, consider 3D rotations around the vertical axis. An eigenvector of this transformation does not change direction, only magnitude. If the unit vector is vertical, it is an eigenvector. If not, the direction of the unit vector changes, so it is not an eigenvector.

Answer 3

If by unit vector you mean a vector of length one, then this is not necessarily an eigenvector of every transformation. E.g. the vector can be rotated by a transformation to point in a different direction (rotation matrix), hence it changed.

Also notice, that an eigenvector is indeed allowed to change during a transformation, but only its length, not its direction. A vector that does not change at all is a fixed point of the transformation, and also a special eigenvector of the transformation, namely an eigenvector to the eigenvalue $\lambda=1$.

Even more, eigenvectors are not determined uniquely. Once you found one, you found infinitely many, because every scaled version of your vector is an eigenvector too. In this sense, every eigenvector can be scaled to be a unit vector, and this unit vector is an eigenvector too. But the reverse is false: not every unit vector is an eigenvector.