Eigenvectors and stretch factors

Parametric equations

Let $A$ be a $2\times 2$ matrix. A nonzero vector $\mathbf{v}\ne 0$ that satisfies

is an eigenvector, and $\lambda$ is the matching eigenvalue. The line through the origin along $\mathbf{v}$ maps to itself. The vector only stretches, reverses, or flattens.

If $\lambda >0$, it keeps its direction and its length is scaled by $\lambda$ (stretched if $\lambda >1$, shrunk if $0<\lambda <1$). If $\lambda <0$, it reverses and is scaled by $|\lambda |$. If $\lambda =0$, that direction is crushed to the origin. When the eigenvalues are complex, the plane has no such line through the origin.

Interaction

• Stretch / Rotation / Shear / Reflection / Saddle / Scalar: switch the matrix and watch the eigen directions and the grid
• u: drag the blue arrow on the figure. Its direction turns, unlike a vector that already lies on an eigenline
• Advanced matrices: show the sliders a, b, c, and d, and the presets Rotate-and-stretch, Flatten, and Oblique. The default view hides the entry sliders
• Reset: return to Stretch

What to notice

• An eigenvector is a direction that still lies on the same line through the origin after the transform. If $\lambda >0$ it is scaled in the same direction, if $\lambda <0$ it reverses, and if $\lambda =0$ it is crushed to the origin.
• Two different real eigenvalues usually give two eigen directions. A repeated eigenvalue may give only one, or every direction may be an eigen direction.
• A pure rotation by an angle that is not a multiple of $\pi$ has no nonzero vector of unchanged direction in the real plane. That matches a rotation moving every direction.

Related

• Linear transformation grid
• Rotation and scaling, composed
• Matrix and linear transformation

Further reading

• Eigenvalues and eigenvectors (Wikipedia)
• Linear map (Wikipedia)