How can I visualize the effect of a transformation from real to complex points?

Question

Is there a convenient or common way to visualize the effect of a matrix multiplication that takes real valued points to complex valued ones.

In particular, if I eigen-decompose a matrix that is easily visualized (as rotations and stretches), I can end up with component matrices that transform points from complex to real.

For example, if I start with

and apply

$$\begin{bmatrix} 0.657 & -1.147 \\ 0.264 & 0.448 \\ \end{bmatrix}$$

I get

and I can easily visualize the intermediate steps of the singular value decomposition:

followed by

But if I preform an eigen-decomposition, the first step ($V^{-1}$) is

$$\begin{bmatrix} 0.543021\, -0.0921334 i & +1.30006 i \\ 0.543021\, +0.0921334 i & -1.30006 i \\ \end{bmatrix}$$

which produces complex points that are not easily visualized in the same way.

Is there a convenient or commonly used way to visualize the effect of a matrix multiplication that takes real valued points to complex valued ones, that is analogous to the real visualizations above (and could be used in their place to present the steps performed)?

Answer 1

I am afraid that, once you leave the real number field, the geometric interpretation of linear mappings is lost. For example, in a complex vector space the vectors $v$ and $iv$ are collinear, but if you try to define a sensible notion of "angle", you will find that the angle between them is $\pi/2$ radians. There is no hope of visualizing such thing in pictures.

After all, you have to pay something in exchange of the benefits that complex numbers give you (all matrices have eigenvalues, etc...).

Answer 2

Loosely we can interpret an eigenvector as a vector that doesn't change direction during the matrix transform. However there is one class of matrices that clearly doesn't have any vectors that stay fixed besides the zero vector, the rotation matrices. There is an interesting connection between certain $2 \times 2$ matrices and the complex numbers which happens to have a useful geometric interpretation that works in both cases.

First define $J:= \begin{bmatrix}0&-1\\1&0\end{bmatrix}$ and consider matrices of the form $aI + bJ$ where $a,b \in \mathbb{R}$ and $I$ is the $2 \times 2$ identity matrix. First, note that $J$ is a rotation matrix that rotates vectors in the plane by $\pi/4$ radians. In fact all the rotation matrices are of the form $aI+bJ$ form with the additional constraint that $a^2+b^2=1$, or equivalently $\det(aI+bJ)=1$. So what about the matrices which aren't of this form? Well, note that since $a,b \in \mathbb{R}$ their squares are positive real numbers so the positive number $r:=\sqrt{a^2 + b^2}$ is uniquely defined and always real. I leave it to you to confirm that $r=0$ if and only if $a=b=0$. So if $r \neq 0$ and $r\neq 1$ then we define $a':= a/r$ and $b':=b/r$. I leave it to you to verify that $aI+bJ=r(a'I+b'J)$ and $\det(a'I+b'J)=1$. This means that matrices of this form have two components, the scaling factor $r$ and the rotation factor $a'I+b'J$. This characterizes matrices of this form. If I multiply two of them together it's a composition of those scalings and rotations, which is again of the same form.

Now consider the complex numbers in the complex plane. We can write any complex number in polar form as $r(\cos \theta + i \sin \theta)$ with $0\leq \theta <2\pi$ and $0\leq r$, the radius. Notice that $(\cos\theta)^2 + (\sin\theta)^2=1$. Now by Euler's Formula we have that $r(\cos \theta + i \sin \theta)=re^{i\theta}$. So if I multiply two complex numbers $(r_1e^{i\theta_1})(r_2e^{i\theta_2})=r_1r_2e^{i(\theta_1+\theta_2)}$ which gives us a convenient geometric interpretation of complex multiplication in the complex plane. The scaling factors $r_1$ and $r_2$ are multiplied and the angles $\theta_1$ and $\theta_2$ are added. In particular note that if I multiply every number in the complex plane by $i$ that it rotates them all by $\pi/4$ radians.

By now you may have guess where I'm going with this. Matrices of the form given can be identified, up to isomorphism, with the complex numbers by looking at the bijective function $\phi:a+bi \rightarrow aI+bJ$. You can check for yourself that these matrices satisfy all the usual conditions of the complex numbers.

But it gets even better. If we look at the characteristic equation of $J$, it's $x^2+1=0$, which is solved by $x=i,-i$. It's eigenvalues agree with our intuition, the rotation matrix matches exactly with the complex numbers that rotate through the same angle with a choice in how you orient the plane. This is because we could just have easily used $-J$ instead of $J$ and the algebra would be identical just as it is for the complex numbers. You can see now that the polar form of a complex number matches with the form $r(a'I+b'J)$ for the matrices mentioned as well.

So loosely, this is how I interpret complex eigenvalues and vectors in a geometric way.