Eigenvalues and eigenvectors

Higher Level

The matrix is [[2, 1], [1, 2]]. The green arrow is a vector, the orange one is where the matrix sends it. Sweep it round and look for the directions where the two line up.

0°

26.6°turned by

2.24×stretched by

not an eigenvectorverdict

Most directions come back pointing somewhere else.

The definition

Mv = λv. The matrix sends v to a multiple of itself: same line, possibly stretched, possibly flipped, but not turned off its direction. λ is the eigenvalue, v the eigenvector.

M(1, 1)= (3, 3) = 3(1, 1) λ = 3 M(1, −1)= (1, −1) = 1(1, −1) λ = 1 M(1, 0)= (2, 1), which is not a multiple of (1, 0)

Finding them

Solve the characteristic polynomial det(M − λI) = 0. For a 2×2 that is λ² − (trace)λ + (det) = 0, where the trace is the sum of the diagonal.

trace= 2 + 2 = 4 det= 4 − 1 = 3 so→ λ² − 4λ + 3 = 0 λ= 3 and 1 check→ 3 + 1 = trace ✓, 3 × 1 = det ✓

That check is free and catches almost every slip. The eigenvalues always sum to the trace and multiply to the determinant. Then substitute each λ back into (M − λI)v = 0 to get its eigenvector.

An eigenvector is a direction, not a single arrow. (1, 1), (2, 2) and (−3, −3) are all the same eigenvector, so any sensible multiple is accepted. Zero is never an eigenvector, because every matrix sends it to itself and it would tell you nothing.

What they are for

They say what a repeated process settles into. Apply a matrix over and over, as a population or a Markov chain does, and the largest eigenvalue decides the long-run growth rate while its eigenvector decides the shape it settles into. That is the same steady state as in 4.19, arrived at from the other side.

Worked example

Bangkok's population splits between the inner districts and the outer ones. Each year the matrix M = [[0.8, 0.3], [0.2, 0.7]] moves people between them.

Find the eigenvalues and eigenvectors, and say what they mean for the city.

trace = 0.8 + 0.7 = 1.5, det = 0.56 − 0.06 = 0.5Both come straight off the matrix and both will check the answer later.
λ² − 1.5λ + 0.5 = 0The characteristic polynomial for a 2 by 2 is always lambda squared minus trace lambda plus determinant.
λ = 1 and λ = 0.5Check: 1 + 0.5 = 1.5 = trace ✓ and 1 × 0.5 = 0.5 = det ✓.
For λ = 1: (0.8 − 1)x + 0.3y = 0, so y = ²⁄₃x, giving (3, 2)Substitute each eigenvalue back and solve. Any multiple of (3, 2) is the same answer.
λ = 1 with (3, 2) is the steady state: 60% inner, 40% outerAn eigenvalue of exactly 1 means a distribution that does not change at all, and (3, 2) normalised is 0.6 and 0.4. The other eigenvalue, 0.5, says any departure from that split halves every year, which is why the city settles so quickly.

Your turn

1. M = [[2, 1], [1, 2]]. Give the larger eigenvalue.

2. The eigenvalues of a 2×2 matrix are 3 and 1. Give its determinant.

3. M(1, 1) = (3, 3). This tells you:

(1, 1) is an eigenvector with eigenvalue 3. (3, 3) is an eigenvector with eigenvalue 1. The determinant is 3. Nothing, without the characteristic polynomial.

Where the marks go

The characteristic polynomial written out before it is solved.

Checking the eigenvalues against the trace and the determinant.

Giving an eigenvector in its simplest whole-number form, and saying what it means when the question is about a population or a process.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

Find the top-left entry of AB where A = [[1,2],[3,4]] and B = [[0,1],[1,0]].

Row by column: 1(0) + 2(1) = 2. Elementwise would give 0, which is the usual first attempt.
Which method?

A commuter chain has transition matrix [[0.8,0.3],[0.2,0.7]]. What proportion uses the bus in the long run?

Steady state: solve Ts = s with the entries summing to 1, giving 0.6. No starting distribution is needed, which is the point.
Which method?

Find the sum to infinity of 4, 2, 1, 0.5, …

Geometric with r = 0.5: 4/0.5 = 8. The partial sums approach it and never arrive.

The same idea elsewhere

4.19, transition matrices: The steady state IS the eigenvector of the transition matrix with eigenvalue 1.
5.17, phase portraits: Eigenvalues decide whether a system settles, spirals or runs away.

Want a verdict on your own draft?

These pages are free and stay free, but they are general and your IA is not. Send me your research question, or whatever exists so far, and I will tell you in writing whether the topic has a ceiling on it, where the marks are going, and what to change first. That costs nothing and it comes back within 24 hours.

Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.

Get a free verdict Full written review, $99

I never write any part of it. Not a sentence, not a calculation, not your data. Under 18: a parent buys this and the thread is with them. I do not work with students at my own school.