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Eigenvectors are special because they do not rotate off their original line (span) during a transformation; they only stretch, shrink, or flip. The zero vector is excluded from this definition because it maps to itself under any linear transformation ($A\vec{0}=\vec{0}$), providing no directional information about the transformation's effect.

Conditions: Vector $\vec{v}$ must be non-zero ($\vec{v} \neq \vec{0}$); Transformation is linear

Eigenvectors are special because they do not rotate off their original line (span) during a transformation; they only stretch, shrink, or flip. The zero vector is excluded from this definition because it maps to itself under any linear transformation ($A\vec{0}=\vec{0}$), providing no directional information about the transformation's effect.

Conditions: Vector $\vec{v}$ must be non-zero ($\vec{v} \neq \vec{0}$); Transformation is linear