Teacher page: Eigenvalues and eigenvectors

The one thing to do with the animation

Sweep the vector right round before writing any algebra down.

Almost every direction comes back pointing somewhere else. Exactly two lie along their own line, and the readout drops to zero degrees turned. That is the definition, discovered.

The characteristic polynomial then has an obvious job: it is how you find those two directions without sweeping a circle by hand.

The free check

Eigenvalues sum to the trace and multiply to the determinant. For [[2,1],[1,2]] that is 3 + 1 = 4 and 3 × 1 = 3. It costs nothing and catches nearly every algebraic slip.

An eigenvector is a direction, not an arrow. (1,1), (2,2) and (−3,−3) are the same answer, so accept any sensible multiple. Zero is never an eigenvector.

The answers

1. Larger eigenvalue	3; the roots of λ² − 4λ + 3.
2. Determinant from 3 and 1	3.
3. M(1,1) = (3,3)	A, (1,1) is an eigenvector with eigenvalue 3.

Where the marks go

1 markThe characteristic polynomial written before it is solved.

1 markEigenvalues checked against trace and determinant.

1 markAn eigenvector in simplest whole-number form, interpreted in context.

What each wrong answer tells you

They give	What it means
4 (Q1)	Gave the trace, which is the sum of the eigenvalues.
1 (Q1)	Gave the smaller one.
4 (Q2)	Confused trace with determinant: sum against product.
B (Q3)	Swapped eigenvector and eigenvalue. The input is the eigenvector.
D (Q3)	Did not recognise Mv = λv as the definition itself.

Other things they will say

"What are they actually for?" The long run. Apply a matrix repeatedly and the largest eigenvalue sets the growth rate while its eigenvector sets the shape it settles into. That is the same steady state as 4.19.

"Can an eigenvalue be negative or zero?" Yes. Negative flips the direction; zero means the matrix collapses that direction entirely, and the determinant is then zero too.

"Why is zero not an eigenvector?" Because every matrix sends it to itself, so it would satisfy the definition for every λ and tell you nothing.

A possible order

	What is happening
1	Sweep the circle. Find the two directions before naming anything.
2	Mv = λv, written against what was just seen.
3	The characteristic polynomial, with the trace and determinant check.
4	Finding the eigenvectors by substituting back.
5	The long-run application, linked to 4.19.

Two things not to say

Do not open with det(M − λI) = 0. It is a method for finding something they have not yet seen the point of.

Do not insist on one particular eigenvector. Any non-zero multiple is the same answer.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

EigenvaluesFind the eigenvalues of [[3,0],[0,5]].
3 and 5
Check an eigenvectorShow that (1, 1) is an eigenvector of [[4,1],[2,3]] and give its eigenvalue.
It maps to (5, 5) = 5(1, 1), so λ = 5. Testing takes one multiplication; assuming is how marks go.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

Full workingFind the eigenvalues and eigenvectors of [[4,1],[2,3]].
λ = 5 with (1,1); λ = 2 with (1,−2). Trace 7 and determinant 10 both check.
Interpret a long runA population matrix has eigenvalues 1.2 and 0.4. What happens to the population over many years, and what sets the shape it settles into?
It grows by about 20% a year, and the eigenvector of 1.2 sets the proportions it settles into.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Work backwardsA 2×2 matrix has eigenvalues 6 and −1. Write down its trace and determinant, then give one matrix with those eigenvalues.
Trace 5, determinant −6. For example [[6,0],[0,−1]], or any matrix similar to it.
Explain a zero eigenvalueWhat does it mean for a matrix to have an eigenvalue of 0?
A whole direction is collapsed to the origin, so the determinant is 0 and the matrix has no inverse.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.