Geometric Interpretation of a Matrix Transformation's Eigen Vectors

Question

I'm having a little bit of an issue grappling how to determine the eigenvalues and eigenvectors of a matrix given that it is a geometric transformation. Take, for instance, the following matrix:
$\begin{bmatrix} 12 & 5 \\ 5 & -12 \end{bmatrix}$ How can I find its eigenvalues/eigenvectors simply by knowing its a reflection-dilation? Likewise, say I had a rotation dilation-- how would I do it then? Thanks for help.

Answer 1

If you know what kind of transformation is (reflection/dilatation/rotation) you can determine one or more eigenvalues and eigenvectors directly.

For example $Av=1\cdot v$ indicate the line $vt$ is a fixed line and if for $w\perp v$ we have $Aw=-1\cdot w$ then A is a reflection with respect to the line $vt$.

In your example $A=\begin{bmatrix} 12 & 5 \\ 5 & -12 \end{bmatrix}$ we find

• $\lambda_1=13 \quad v_1=(5,1)$
• $\lambda_1=-13 \quad v_2=(-1,5)$

is indeed a reflection with dilatation with respect to the line $v_1t$.

Answer 2

Frankly, the generic computation so simple and quick for $2\times2$ matrices that I’m not sure that there’s much to be gained by memorizing a bunch of special cases. On the other hand, understanding the eigensystems of various types of transformations certainly isn’t a bad thing.

There are, of course, the properties of eigensystems of square matrices of any order. Here are a few:

• The last eigenvalue comes for “free” because the sum of the eigenvalues is equal to the trace of the matrix.
• Real symmetric matrices have real eigenvalues with orthogonal eigenspaces.
• The only eigenvalues of a projection are $0$ and $1$; it’s the identity map when restricted to its image.
• The only eigenvalues of a reflection are $\pm1$. The eigenspace of $1$ is the subspace that corresponds to the reflector.
• The eigensystem of $aI+b\mathbb 1$ has a particularly simple structure explained many times on MSE.
• A dilation is a multiple of the identity, so has a single eigenvalue—the dilation factor—with the entire space as its eigenspace.

A few $2\times2$ specifics:

• For a rotation matrix $R$, $\operatorname{tr}R = 2\cos\theta$ and its eigenvalues are $e^{\pm i\theta}$ with eigenvectors $(\pm i,1)^T$, respectively.
• The eigenvalues of a conformal matrix $\begin{bmatrix}\alpha&-\beta\\\beta&\alpha\end{bmatrix}$ are $\alpha\pm i\beta$. (Indeed, these matrices are isomorphic to the complex numbers under multiplication.) If you convert these complex eigenvectors into the polar form $re^{i\theta}$, you can see that these matrices correspond to dilated rotations.
• A reflection in the line $x\sin\theta-y\cos\theta=0$ has the form $\begin{bmatrix}\cos{2\theta}&\sin{2\theta}\\\sin{2\theta}&-\cos{2\theta}\end{bmatrix}$. Its eigenvectors are $(\cos\theta,\sin\theta)^T$ with eigenvalue $1$ and $(\sin\theta,-\cos\theta)^T$ with eigenvalue $-1$.

You can combine some of these, too. Your example of a dilation/reflection must have the form $$\begin{bmatrix}s\cos{2\theta}&s\sin{2\theta}\\s\sin{2\theta}&-s\cos{2\theta}\end{bmatrix}$$ and its eigenvalues are $\pm s$. You can recover $s$ using the identity $(s\cos{2\theta})^2+(s\sin{2\theta})^2=s^2$, in this case producing $s = \sqrt{12^2+5^2}=13$. You could compute the eigenvectors via the half-angle trigonometric formulas, but as I mentioned at the start the generic method is pretty fast and spits out the eigenvectors $(5,1)^T$ and $(1,-5)^T$ in fairly short order.

For your dilation/rotation, see the case of a conformal matrix, above. Taking, for example, $$\begin{bmatrix}12&5\\-5&12\end{bmatrix},$$ you can immediately write down the eigenvalues $12\pm5i$, and the dilation factor is again $13$.