5.3 Complex Eigenvalues and the Spiral

Some matrices turn every direction. The cleanest examples are the rotation–scaling matrices: rotate the plane by an angle $b$ , then scale it by a factor $a greater than 0$ . Written out, this map sends $(x, y)$ to $(a \cos b x - a \sin b y, a \sin b x + a \cos b y)$ . If $sin b not equal to 0$ , every nonzero vector is rotated by the full angle $b$ — no line is invariant, no real eigenvector exists, and the hunts of the last two sections come home empty. The misalignment function never touches zero.

Yet the matrix still has structure, and the way to see it is to iterate. Start at the point $open parenthesis 2, 0 close parenthesis$ and apply the matrix over and over. Each application multiplies the distance from the origin by $a$ and advances the angle by $b$ , so after $k$ steps the point sits at $2 a to the power k open parenthesis cos k b, sin k b close parenthesis$ . Letting the step count run continuously traces the curve $2 a to the power t open parenthesis cos b t, sin b t close parenthesis$ — the violet spiral in the figure, with the iteration's start marked at $open parenthesis 2, 0, 0 close parenthesis$ .

A smooth violet spiral in the horizontal plane, starting at a marked point and winding around the origin. A slider named a sets whether the spiral tightens inward (a below 1), stays a circle (a equal to 1), or unwinds outward (a above 1), while a slider named b sets how fast it wraps around. — The trajectory $2 a to the power t open parenthesis cos b t, sin b t close parenthesis$ for $t$ from $0$ to $40$ , starting at the marked point $open parenthesis 2, 0, 0 close parenthesis$ . The slider $a$ is the growth factor per unit time and $b$ is the rotation rate in radians per unit time; the defaults $a equals 0.96$ , $b equals 0.6$ spiral gently inward.

Explore the figure

At the defaults, follow the curve from the marked start point. With $b equals 0.6$ it wraps around the origin about $3.8$ times over the run, and with $a equals 0.96$ each unit of time shaves four percent off the radius, ending near radius $0.4$ . Rotation and decay, superimposed.
Set $a equals 1$ . The spiral closes into a perfect circle of radius $2$ : pure rotation, nothing gained or lost. This is the knife-edge between decay and growth.
Push $a$ to $1.04$ . The same curve now unwinds outward, finishing near radius $9.6$ . A four-percent difference per step, compounded forty times, is the whole difference between collapse and explosion.
Return $a$ to $0.96$ and sweep $b$ across its range. At $b equals 0.2$ the path barely completes one lap; at $b equals 1.2$ it wraps nearly eight times. Notice what $b$ never changes: the radius at each time depends on $a$ alone.

Eigenvalues without eigenvectors

The eigenvalue machinery does not break here — it reports complex numbers. The characteristic polynomial of the rotation–scaling matrix is

$λ^{2} - 2 a \cos b λ + a^{2} = 0$

and its discriminant is $4 a^{2} \cos^{2} b - 4 a^{2} = - 4 a^{2} \sin^{2} b$ , which is negative whenever $sin b not equal to 0$ : no real roots, confirming what the pictures showed. The roots are the complex pair $λ = a \cos b \pm i a \sin b$ , better read in polar form as $λ = a e^{\pm i b}$ . The two sliders of the figure are literally the two coordinates of the eigenvalue: $a$ is its modulus and $b$ is its argument. A complex eigenvalue does not mark an invariant line; it prescribes a motion — turn by the argument, scale by the modulus, once per application — and the spiral is that prescription traced out in time. The trichotomy you explored is the modulus test: $| λ | less than 1$ spirals in, $| λ | equals 1$ circles, $| λ | greater than 1$ spirals out.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Its chapter sequence is aligned to Interactive Linear Algebra (Margalit & Rabinoff, Georgia Tech, GNU FDL); this work is original, copies nothing from it, and is not affiliated with or endorsed by its authors. License: CC-BY-NC-SA-4.0.