Eigenvalues, Eigenvectors and Diagonalisation

Active learning helps students grasp the geometric meaning of complex numbers by connecting algebraic representations to visual plots. Working with the Argand diagram builds intuition for modulus as distance and argument as angle, making abstract concepts concrete through hands-on tasks and collaborative problem-solving.

• JC 2
• Further Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Calculate the modulus and argument of a complex number given in Cartesian form.
2. Represent complex numbers geometrically on an Argand diagram, identifying their position based on modulus and argument.
3. Convert complex numbers between Cartesian form (a + bi) and modulus-argument form (|z|(cosθ + i sinθ)).
4. Explain the geometric meaning of multiplication and division of complex numbers in terms of rotation and scaling on the Argand diagram.

Ready-to-Use Activities

Maker Learning

⏱ 30 min·Pairs

Pairs Plotting: Modulus and Argument Calculation

Pairs receive cards with complex numbers in Cartesian form. They plot each on a printed Argand grid, draw the radius vector, measure modulus with a ruler, and argument with a protractor while noting quadrants. Partners verify calculations and explain one conversion to each other.

Prepare & details

What do eigenvalues and eigenvectors represent geometrically?

Facilitation Tip: During Pairs Plotting, circulate to ensure students label axes clearly and measure angles from the positive real axis, not from adjacent axes.

Setup: Maker tables with tools and supplies

Materials: Challenge prompt with constraints, Materials inventory sheet, Sketch sheet, Build log, Test record with iteration notes

• Apply
• Analyze
• Create
• Self-Management
• Decision-Making

Maker Learning

⏱ 40 min·Small Groups

Small Groups: Argand Rotation Challenge

Groups plot a complex number, then rotate it by given angles using protractors to find new arguments. They convert results back to Cartesian form and check with calculators. Share one rotation example with the class.

Prepare & details

How is the characteristic equation used to find eigenvalues?

Facilitation Tip: For the Argand Rotation Challenge, provide protractors and rulers so groups can verify angle additions and length changes through physical measurement.

Setup: Maker tables with tools and supplies

Materials: Challenge prompt with constraints, Materials inventory sheet, Sketch sheet, Build log, Test record with iteration notes

• Apply
• Analyze
• Create
• Self-Management
• Decision-Making

Maker Learning

⏱ 25 min·Whole Class

Whole Class: Desmos Interactive Exploration

Project a Desmos graph of the Argand plane. Call on students to suggest complex numbers; compute and plot modulus-argument live as a class. Discuss patterns in rotations and scalings visible on screen.

Prepare & details

Why is matrix diagonalisation useful in computational mathematics?

Facilitation Tip: In the Desmos Interactive Exploration, pause the class after each prompt to discuss observations as a whole group to reinforce key insights.

Setup: Maker tables with tools and supplies

Materials: Challenge prompt with constraints, Materials inventory sheet, Sketch sheet, Build log, Test record with iteration notes

• Apply
• Analyze
• Create
• Self-Management
• Decision-Making

Maker Learning

⏱ 35 min·Individual

Individual: Mod-Arg Conversion Relay

Students work individually on a sheet converting 10 complexes both ways, then pass to a partner for spot-checks. Circulate to prompt quadrant reasoning. Debrief common issues as a class.

Prepare & details

What do eigenvalues and eigenvectors represent geometrically?

Facilitation Tip: For the Mod-Arg Conversion Relay, check that students record both the converted form and the original number to build connection between representations.

Setup: Maker tables with tools and supplies

Materials: Challenge prompt with constraints, Materials inventory sheet, Sketch sheet, Build log, Test record with iteration notes

• Apply
• Analyze
• Create
• Self-Management
• Decision-Making

Teaching This Topic

Teach this topic by prioritizing visual and kinesthetic experiences before formal notation. Start with plotting to build geometric intuition, then introduce modulus and argument through measurement rather than immediate formula memorization. Avoid rushing to polar form before students can confidently interpret the diagram, and use physical tools like rulers and protractors to ground abstract concepts in tangible experience.

What to Expect

Students will confidently plot complex numbers, calculate modulus and argument correctly, and explain how rotation and scaling affect these values. They will also recognize the equivalence between Cartesian and modulus-argument forms through direct measurement and transformation activities.

Watch Out for These Misconceptions

Common Misconception

During Pairs Plotting, watch for students who assume the argument is always between 0 and 90 degrees.

What to Teach Instead

Have pairs plot points in all four quadrants and measure angles directly, then compare their measurements with calculator outputs to recognize the full range of possible arguments.

Common Misconception

During the Argand Rotation Challenge, watch for confusion that modulus changes sign based on quadrant.

What to Teach Instead

Provide rulers and have groups measure the distance from the origin for points in different quadrants, observing that length remains positive regardless of position.

Common Misconception

During the Mod-Arg Conversion Relay, watch for the belief that converting to mod-arg form erases the real-imaginary structure.

What to Teach Instead

Ask students to expand their mod-arg results back to Cartesian form using the angle addition formulas, verifying that the original real and imaginary parts reappear.

Assessment Ideas

Quick Check

After Pairs Plotting, give students 3-4 complex numbers in Cartesian form and ask them to plot each on a mini-Argand diagram and calculate its modulus and principal argument. Collect their diagrams and calculations to check for correct plotting and computation.

Discussion Prompt

During the Argand Rotation Challenge, pose the question: 'How does multiplying two complex numbers relate to their individual moduli and arguments?' Listen for responses that include the product of moduli as the new modulus and the sum of arguments as the new argument, linking these to rotation and scaling.

Exit Ticket

After the Mod-Arg Conversion Relay, give students a complex number in modulus-argument form and ask them to convert it to Cartesian form and sketch its position on an Argand diagram. Review both the algebraic conversion and the geometric representation for accuracy.

Extensions & Scaffolding

• Challenge students to find three different complex numbers that multiply to produce a given target in modulus-argument form, requiring them to combine rotations and scalings creatively.
• For students who struggle, provide pre-labeled grids with marked quadrants and offer step-by-step prompts to calculate argument with quadrant checks built in.
• Allow extra time for students to explore how changing the argument by π/2 affects a complex number’s position, linking to multiplication by i and rotations.

Key Vocabulary

Argand Diagram	A graphical representation of complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Modulus	The distance of a complex number from the origin on the Argand diagram, calculated as |z| = sqrt(a² + b²) for z = a + bi.
Argument	The angle formed by the line segment connecting the origin to the complex number and the positive real axis on the Argand diagram, denoted by θ.
Modulus-Argument Form	A way of expressing a complex number as z = |z|(cosθ + i sinθ), where |z| is the modulus and θ is the argument.