# symmetry-eigenbasis-stretch.mp4

Chapter section: Primer -- Eigenfunctions

Caption (verbatim):

Eigenvectors pictured

Image description (verbatim):

Stretching in ordinary and eigenvector bases

## Before the animation

Chapter lines 614–616:

| $(x,y)$ basis: components mix | $(u,v)$ basis: components do not mix |
|---|---|
| $\displaystyle \begin{aligned}\mathbf r&=\frac12\begin{pmatrix}s+s^{-1}&s-s^{-1}\\s-s^{-1}&s+s^{-1}\end{pmatrix}\mathbf r_{\mathrm{in}}\\&=\frac12\left[(s+s^{-1})x+(s-s^{-1})y\right]\hat{\mathbf x}\\&\quad+\frac12\left[(s-s^{-1})x+(s+s^{-1})y\right]\hat{\mathbf y}\end{aligned}$ | $\displaystyle \begin{aligned}\mathbf r&=\begin{pmatrix}s&0\\0&s^{-1}\end{pmatrix}\mathbf r_{\mathrm{in}}\\&=su\hat{\mathbf u}+s^{-1}v\hat{\mathbf v}\end{aligned}$ |

Chapter lines 618–618:

This is readily understood visually:

## After the animation

Chapter lines 626–626:

Now for a bunch of terminology. The "natural" basis vectors are the **eigenvectors**, the values a transformation scales these by are the **eigenvalues** and the basis they form is called the **eigenbasis**. When the "vectors" are functions, we call them **eigenfunctions**. As the transformation matrix is diagonal in the eigenbasis, the procedure for finding an eigenbasis is typically called **diagonalization**.