Activity: Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

Quantum Fundamentals 2021
Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
What students learn
  • Linear transformations can act on complex-valued vectors;
  • Eigenvalues can be complex;
  • Linear transformations can have complex eigenvectors even when they have no real eigenvectors.

Instructor's Guide

Introduction

Students use their left arm to represent a complex number in the complex plane (Argand diagram) where their left shoulder is the origin. They practice using this representation to explore various transformations of two-component complex vectors. The same convention should be chosen across the classroom; typically the partner on the left side represents the top component, and the one on the right side represents the bottom component.

Student Conversations

Each example is chosen to highlight a particular aspect of the representation.
  • Representing two component vectors in pairs, where one person is the top component and the other is the bottom component. These simple examples help to orient the students to representing two component vectors in pairs. \[\pmatrix{1\\i} \rm{and} \pmatrix{1\\-i}\]
  • Transformations of two component vectors. This example highlights the idea that what is meant by an eigenvalue is that both components have the same thing happen to them and that if the eigenvalue is complex, it means a rotation in the complex plane. \[e^{i\pi/4}\pmatrix{1\\i}=\pmatrix{e^{i\pi/4}\\ie^{i\pi/4}}=\pmatrix{e^{i\pi/4}\\e^{i3\pi/4}}\]
  • This provides an example of a transformation where different things happen to each component. \[\pmatrix{i&0\\0&1}\pmatrix{1\\i}=\pmatrix{i\\i}\]
  • This is an example of a rotation, a transformation where both components stay real and the total length stays constant. The relative magnitude of the two components depends on the value of \(\theta\). \[\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\0}=\pmatrix{\cos\theta\\-\sin\theta}\] \[\rightarrow\dfrac{1}{\sqrt{2}}\pmatrix{1\\-1}\] when \(\theta=\dfrac{\pi}{4}\)
  • This next example is the same rotation, now acting on a two-component complex vector and can be used to highlight the connection between transformations and eigenvalues and eigenvectors. \[\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\i}=\pmatrix{\cos\theta+i\sin\theta\\-\sin\theta+i\cos\theta}=\dfrac{1}{\sqrt{2}}\pmatrix{1+i\\-1+i}=\pmatrix{e^{i\pi/4}\\e^{i3\pi/4}}=e^{i\pi/4}\pmatrix{1\\i}\]

Wrap-up

After each example, a short wrap-up should be done by having one or more groups discuss what they did and why. Highlight the relative features of the particular example.
Author Information
Corinne Manogue
Keywords
arms complex numbers phase rotation reflection math
Learning Outcomes