Complex Numbers and Oscillatory Modes

Extend the number system

Define $i^2=-1$ . A complex number is $z=a+ib$ , with conjugate $\bar z=a-ib$ and magnitude $|z|=\sqrt{a^2+b^2}$ . Add components and multiply using $i^2=-1$ . To divide by nonzero $z$ , multiply numerator and denominator by its conjugate.

Polar form is $z=r(\cos\theta+i\sin\theta)=re^{i\theta}$ . Multiplication multiplies magnitudes and adds angles, so $z^k=r^ke^{ik\theta}$ . This identity follows from the angle-addition formulas and explains rotation combined with scaling.

Visual guide

VISUAL GUIDEComplex eigenvalues can describe real rotation

Unit circle

The real matrix [[0, −1], [1, 0]] rotates vectors by a quarter turn. The four arrows show repeated application to (1, 0). No nonzero real direction is preserved, consistent with eigenvalues ±i.

Worked example: arithmetic and roots

$(1+2i)(3-i)=5+5i$ . Also $1/(1+i)=(1-i)/2$ . The equation $\lambda^2+1=0$ has roots $i$ and $-i$ ; there is no real solution because a real square cannot be negative.

For real matrices, nonreal eigenvalues occur in conjugate pairs. Conjugating $Av=\lambda v$ gives $A\bar v=\bar\lambda\bar v$ since $A$ itself is real.

Worked example: a real rotation

Let $R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ . Its eigenvalues are $\pm i$ . For $\lambda=i$ , a complex eigenvector is $(1,-i)^T$ . This does not mean the original real plane contains a fixed real direction: a quarter-turn changes every nonzero real direction.

Nevertheless real motion is easy to interpret. Identifying $(x,y)$ with $z=x+iy$ , applying $R$ multiplies $z$ by $i$ , a $90^\circ$ counterclockwise rotation. For $A=\rho R$ , repeated application multiplies magnitudes by $\rho^k$ . With $0<\rho<1$ , states spiral toward zero; with $\rho>1$ , they spiral outward.

Complex vector inner products use conjugate transpose $u^*v$ , rather than ordinary transpose, to ensure $v^*v$ is real and nonnegative. Our earlier projection formulas were stated for real vectors.

Practice

Find $|3-4i|$ and its conjugate.
Compute $i^4$ .
Describe repeated action of $0.8R$ on a nonzero real vector.

Show worked solutions

Magnitude $5$ ; conjugate $3+4i$ .
$i^2=-1$ , so $i^4=1$ .
Rotate by $90^\circ$ each step and shrink magnitude by $0.8$ ; states tend to zero.