This video,

[Source: YouTubediscussion]

describes the mathematics behind one of Richard Feynman’s lectures on the elliptical orbits of planets.

Interestingly, I later encountered the same (conceptual) description at about the 3-minute mark in this lecture, on the Fourier transform (“An animated introduction to the Fourier Transform, winding graphs around circles.”),

[Source: YouTube]

This page, Understanding the Fourier Transform  [mirror] describes the Fourier transform very succinctly.


[Image source. Click image to open in new window.]

That figure is explained in the accompanying blog posts.

Basically, whenever you see $\small i$ and $\small \pi$, think “circles and rotation.”

Euler’s formula ties in with $\small i$ and rotation and eigenvectors, as discussed in the really excellent blog post Intuitive Understanding of Euler's Formula:

Euler’s formula also comes up in my background discussion on graph signal processing. These complex numbers manifest later in that discussion, when I use GNU Octave to determine some eigenvalues in my various examples: I get “complex” eigenvectors of the form $\small 0.14037 + 0.55581i$  (which is reminiscent of Euler’s formula, $e^{i\phi} = cos\phi + (i)sin \phi$):

octave:>> c = [5,8,16; 4,1,8; -4,-4,-11]
c =
    5    8   16
    4    1    8
   -4   -4  -11

octave:>> [eigenvectors, eigenvalues] = eig(c)
eigenvectors =
   0.81650 + 0.00000i   0.14037 - 0.55581i   0.14037 + 0.55581i
   0.40825 + 0.00000i  -0.71521 + 0.00000i  -0.71521 - 0.00000i
  -0.40825 + 0.00000i   0.28742 + 0.27790i   0.28742 - 0.27790i

eigenvalues =
Diagonal Matrix
   1.00000 + 0.00000i                    0                    0
                    0  -3.00000 + 0.00000i                    0
                    0                    0  -3.00000 - 0.00000i

octave:>> eig(c)
ans =
   1.00000 + 0.00000i
  -3.00000 + 0.00000i
  -3.00000 - 0.00000i


Stated in [Imaginary Numbers] Complex Eigenvalues,

    "Complex numbers are added like vectors:
      $\small x + iy + u + iv = (x + u) + i(y + v)$
    and multiplied as
      $\small z * w = (x + iy)(u + iv) = xu - yv + i(yu - xv)$.
    If $\small z \neq 0$, one can divide
      $\small 1/z = 1/(x + iy) = (x - iy)/(x^2 + y^2)$."

… hinting (as we already know) at the relationship between vectors (e.g. vector space, models) and imaginary numbers. 

That same source includes these statements,

    "Geometric interpretation. If $\small z = x + iy$ is written as a vector $\small \begin{align} \begin{bmatrix}x \\ y\end{bmatrix} \end{align}$, then multiplication with an $\small y$ other complex number $\small w$ is a dilation-rotation: a scaling by $\small |w|$ and a rotation by $\small arg(w)$."

    "Unit circle. Complex numbers of length $\small 1$ have the form $\small z = e^{i\phi}$ and are located on the unit circle. The characteristic polynomial $\small f_A(\lambda) = \lambda^5 - 1$ of the matrix
      $\small \begin{align} \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 0 & 1 \\\ 1 & 0 & 0 & 0 & 0 \end{bmatrix} \end{align}$
    has all roots on the unit circle. The roots $\small e^{2 \pi ki/5}$, for $\small k = 0, \ldots, 4$ lie on the unit circle".

These concepts are elaborated in Geometry of the Eigenvectors of Plane Rotation, which states the following.  [A good background before reading this material is Intuitive Understanding of Euler's Formula,]

    "Rotation of a two-dimensional vector $\small (x,y)$ in the Cartesian plane can be expressed as a matrix multiplication, where the rotated vector $\small (x',y')$ is equal to the product of the rotation matrix $\small M$ and the original (column) vector $\small (x,y)$. For rotation by a counter-clockwise angle $\small A$, the antisymmetric $\small 2x2$ matrix $\small M$ has the cosine of $\small A$ in the diagonal elements, the sine of $\small A$ at lower left, and the negative of the sine of $\small A$ at upper right. ...

    "For an arbitrary angle $\small A$, however, $\small M$ has the two complex-conjugate eigenvalues $\small c = e^{iA}$ and $\small c^* = e^{-iA}$. These complex eigenvalues have meaning only if the setting is generalized to allow $\small x$ and $\small y$ to be complex, so that the products of $\small c$ and $\small x$ and of $\small c$ and $\small y$ are again elements of the vector space. The product of $\small c$ and a complex number $\small z$ is itself equivalent to a rotation of $\small z$ in the complex plane by the angle $\small A$. To see this, represent $\small z$ in polar coordinates as magnitude times exponential of the product of $\small i$ and the angle; then multiplying by $\small c$ just adds $\small A$ to the angle. Thus, multiplication of the complex eigenvector $\small (x,y)$ by the eigenvalue $\small c$ or $\small c^*$ is equivalent to rotation of the complex $\small x$ and $\small y$ by the angle $\small A$ (for $\small c$) or the angle $\small -A$ (for $\small c^*$) in the respective complex $\small x$ and $\small y$ planes.

    "The matrix $\small M$, however, is real. Thus, for complex $\small x$ and $\small y$, the product of $\small M$ and $\small x$ or $\small y$ can be viewed as the product of $\small M$ and the real vector $\small Re(x,y)$ plus $\small i$ times the product of $\small M$ and the real vector $\small Im(x,y)$, where $\small Re$ and $\small Im$ denote the real and imaginary parts, respectively. Each of these real products is a rotation by $\small A$ in a real two-dimensional plane: the first is the rotation by $\small A$ in the plane of the real parts of $\small x$ and $\small y$, and the second is the rotation by $\small A$ in the plane of the imaginary parts of $\small x$ and $\small y$. Each of these is equivalent in form to the original rotation of the original real vector $\small (x,y)$.

    "The eigenvalue condition in the complex setting can therefore be viewed as a commutation relation: rotation by $\small A$ of the vector of real parts of $\small x$ and $\small y$ in the plane $\small Im(x,y)=(0,0)$, followed by rotation by $\small A$ of the vector of imaginary parts of $\small x$ and $\small y$ in the plane $\small Re(x,y)=0$, gives the same result as rotation of the complex $\small x$ by $\small A$ (or $\small -A$, for $\small c^*$) in the plane $\small y=0$, followed by rotation of the complex $\small y$ by $\small A$ (or $\small -A$, for $\small c^*$) in the plane $\small x=0$. ...

    "The vector space in the complex setting is effectively four dimensional, with independent real and imaginary parts for each of $\small x$ and $\small y$. This makes the geometry of the rotations by $\small M$ and $\small c$ or $\small c^*$ difficult to depict or visualize. ..."
    [Image source. Click image to open in new window.]

Sources and Additional Reading

Imaginary Numbers