Eigenvalues and Eigenvectors

IB Mathematics Applications and Interpretation HL· Topic 1.15: Eigenvalues and eigenvectors for 2x2 and 3x3 matrices· 12 min read

1. Core Definition of Eigenvalue-Eigenvector Pairs★★☆☆☆HL only⏱ 3 min

For any n x n square matrix A, an eigenvector ( \boldsymbol{v} ) is a non-zero vector that, when multiplied by A, returns a scalar multiple of itself. This scalar multiple is the corresponding eigenvalue λ.

📘 Definition

Eigenvalue-Eigenvector Identity

A\boldsymbol{v} = \lambda \boldsymbol{v}

The core identity that defines the relationship between matrix A, its eigenvalue λ, and non-zero eigenvector v

Example:

For matrix ( A = \begin{pmatrix} 2 & 0 \ 0 & 3 \end{pmatrix} ), ( \lambda = 2 ) and ( \boldsymbol{v} = \begin{pmatrix}1 \ 0\end{pmatrix} ) satisfy the identity

📐 Worked Example

Verify that ( \boldsymbol{v} = \begin{pmatrix} 2 \ 1 \end{pmatrix} ) is an eigenvector of ( A = \begin{pmatrix} 3 & 2 \ 1 & 4 \end{pmatrix} ) and find its corresponding eigenvalue

1
Calculate the product A v directly
2
$A\boldsymbol{v} = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 8 \\ 6 \end{pmatrix}$
3
Rewrite the resulting vector as a scalar multiple of the original v
4
$\begin{pmatrix} 8 \\ 6 \end{pmatrix} = 4 \begin{pmatrix} 2 \\ 1 \end{pmatrix}$
5
Confirm the scalar multiple is the eigenvalue: λ = 4

✓ Quick check

Test your understanding of the core identity:

Which of the following vectors is an eigenvector of ( \begin{pmatrix} 1 & 0 \ 0 & -2 \end{pmatrix} )?
- ( \begin{pmatrix} 1 \ 1 \end{pmatrix} )
- ( \begin{pmatrix} 0 \ 1 \end{pmatrix} )
- ( \begin{pmatrix} 2 \ 2 \end{pmatrix} )
Reveal answer
$ \begin{pmatrix} 0 \\ 1 \end{pmatrix} $ —
Multiplying this vector by the matrix returns -2 times the original vector, satisfying the eigenvector identity

2. Deriving the Characteristic Equation★★★☆☆HL only⏱ 3 min

Rearranging the core eigenvector identity allows us to derive a polynomial equation whose roots are all eigenvalues of matrix A. This derivation relies on the property that a non-zero vector v can only satisfy ( (A - \lambda I) \boldsymbol{v} = 0 ) if the matrix ( A - \lambda I ) is singular (determinant = 0).

🔬 Derivation

Goal:

Derive the characteristic equation for square matrix A

Starting from:

A\boldsymbol{v} = \lambda \boldsymbol{v}

1
Subtract ( \lambda \boldsymbol{v} ) from both sides: ( A\boldsymbol{v} - \lambda \boldsymbol{v} = 0 )
2
Factor out the identity matrix I to group terms: ( (A - \lambda I) \boldsymbol{v} = 0 )
3
For non-zero v to exist, the matrix ( A - \lambda I ) cannot be invertible, so its determinant equals 0
4
This gives the characteristic equation: ( \det(A - \lambda I) = 0 )

Result:

The roots of this nth-degree polynomial are all eigenvalues of the n x n matrix A

📐 Worked Example

Find the characteristic equation for 2x2 matrix ( A = \begin{pmatrix} 5 & 2 \ 2 & 5 \end{pmatrix} )

1
Construct the matrix ( A - \lambda I )
2
$A - \lambda I = \begin{pmatrix} 5 - \lambda & 2 \\ 2 & 5 - \lambda \end{pmatrix}$
3
Calculate the determinant of this matrix
4
$\det(A - \lambda I) = (5 - \lambda)^2 - 4 = \lambda^2 -10 \lambda +21$
5
Set determinant equal to 0 to get the characteristic equation
6
$\lambda^2 - 10 \lambda + 21 = 0$

3. Calculating Eigenvectors from Eigenvalues★★★★☆HL only⏱ 4 min

Once you have solved the characteristic equation for all eigenvalues, substitute each eigenvalue back into the system ( (A - \lambda I) \boldsymbol{v} = 0 ) to solve for the corresponding eigenvector. All non-zero scalar multiples of a valid eigenvector are also valid eigenvectors for that eigenvalue.

📐 Worked Example

Find all eigenvectors for matrix ( A = \begin{pmatrix} 5 & 2 \ 2 & 5 \end{pmatrix} ) whose characteristic equation is ( \lambda^2 - 10 \lambda + 21 = 0 )

1
Solve the characteristic equation to get eigenvalues λ = 3 and λ =7
2
Substitute λ=3 into ( (A - \lambda I) \boldsymbol{v} = 0 )
3
$\begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = 0 \implies 2v_1 + 2v_2 = 0 \implies v_1 = -v_2$
4
Simplify to get eigenvector for λ=3: ( k \begin{pmatrix} 1 \ -1 \end{pmatrix} ) where k ≠ 0
5
Substitute λ=7 into the system
6
$\begin{pmatrix} -2 & 2 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = 0 \implies -2v_1 + 2v_2 = 0 \implies v_1 = v_2$
7
Simplify to get eigenvector for λ=7: ( m \begin{pmatrix} 1 \ 1 \end{pmatrix} ) where m ≠ 0

4. Exam-Focused Eigenvalue Properties★★★☆☆HL only⏱ 2 min

The sum of all eigenvalues of matrix A equals the trace of A (sum of diagonal elements)
The product of all eigenvalues of matrix A equals the determinant of A
A matrix with a zero eigenvalue is singular (non-invertible)
Eigenvalues of a diagonal matrix are exactly the elements on its main diagonal

5. Common Pitfalls

Wrong move:

Forgetting to subtract λI before calculating the determinant of A

Why:

Produces a non-constant polynomial that is not a valid characteristic equation

Correct move:

Explicitly write out the full A - λI matrix before expanding the determinant

Wrong move:

Substituting the eigenvector back into the original matrix A instead of A - λI when solving

Why:

Leads to an over-constrained system that cannot be solved correctly

Correct move:

Always use the (A - λI) v = 0 system to find eigenvectors

Wrong move:

Submitting a zero vector as an eigenvector

Why:

Eigenvectors are defined as non-zero by the IB syllabus, so zero vectors get zero marks

Correct move:

Explicitly state that the scalar multiple constant cannot equal zero

Wrong move:

Miscalculating the sign of the characteristic polynomial when expanding the determinant

Why:

Leads to incorrect eigenvalues that do not satisfy the trace and product checks

Correct move:

Verify your eigenvalues by checking their sum equals the trace of A

Wrong move:

Using decimal approximations for eigenvalues instead of exact radical forms

Why:

IB exam markers penalize approximate answers for exact eigenvalue calculation questions

Correct move:

Leave eigenvalues in exact surd form unless explicitly told to round

6. Quick Reference Cheatsheet

Task	2x2 Matrix Workflow	3x3 Matrix Workflow	Quick Check
Find characteristic equation	Expand det(A - λI) to get quadratic polynomial	Expand det(A - λI) to get cubic polynomial	Sum of roots equals trace of A
Solve for eigenvalues	Factor or use quadratic formula	Use rational root theorem then factor quadratic	Product of roots equals det(A)
Find eigenvectors	Solve 1 linear equation for ratio of components	Solve homogeneous 3-variable system	Confirm A v = λ v for your pair
Simplify eigenvector	Reduce to smallest integer components	Reduce to smallest integer components	Any non-zero multiple is valid

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · Paper 2
2x2 eigenvalue calculation
2024 · Paper 3
Eigenvector for population model
2023 · Paper 1
Verify given eigenvector pair

What's Next

Mastering eigenvalues and eigenvectors is the critical foundation for advanced AI HL topics that appear frequently in Paper 3 problem sets. You will next apply these skills to diagonalize matrices, which simplifies raising matrices to large integer powers for iterative model calculations. These tools are also core to analyzing Markov chain steady states, a heavily weighted applied topic for population dynamics and predictive modelling assessments. You will also revisit eigenvectors when studying geometric transformations of 2D and 3D shapes, where eigenvectors define lines of invariant direction under transformation.