eigenvector

A nonzero vector $A$ only stretches or shrinks, never turns off its own line.

For an $n\times n$ matrix $A$ , a nonzero vector $\mathbf v$ is an eigenvector of $A$ if $A\mathbf v$ is a scalar multiple of $\mathbf v$ itself: $A\mathbf v = \lambda\mathbf v$ . Multiplying $\mathbf v$ by $A$ never leaves the line $\mathbf v$ spans — no rotation, no shear, only stretching, shrinking, or flipping along that same line. The zero vector is explicitly excluded, since it would satisfy the equation for every $\lambda$ at once.