Eigenvector definition and negative eigen values

Question

A usual definition for eigenvectors are the "Almost all vectors change direction when multiplied by a matrix $A$. Certain exceptional vectors $x$ are in the same direction as $Ax$. These are called eigenvectors." (definition in Strang book).

My question is when the eigenvalue is negative, it means the vector is in the opposite direction right ?. It means the vector did change the direction and the new direction is 180 degrees of the original vector. Am I wrong ?

Answer 1

Think of $Ax=\lambda x$ as meaning the effect of the transformation $A$ on the vector $x$ is equivalent to a scalar multiplication of the vector. Vectors for which this is true under the transformation are known as eigenvectors and the corresponding scalar called eigenvalues.

You have described a situation where $\lambda < 0$ and so the vector points in the opposite direction.

Answer 2

You are correct. If $AX=\lambda X$ where$ \lambda <0$, then $AX$ is in the oposite direction of X but still on the same line.

Things get worse if you have a complex eigenvalue and a complex eigenvector where you can not even say, they stay on the same line.